640 L. Bergström
In view of the question whether or not nonlinear feedback occurs, the presence of nonlinear -terms in the 8-equations deserves some comment. In plane Poiseuille flow, Benney and Gustavsson [28] and Shanthini
[29] have shown that the nonlinear terms of the equation governing the normal velocity v do not contain the transiently amplified normal vorticity η if oblique disturbances of the form e
imαx+βz
are considered α and β are streamwise and spanwise wavenumbers. If instead for example a pair of oblique disturbances of the form
e
iαx±βz
are studied the situation becomes different and nonlinear terms containing η occur in the equation governing v and thus there is a potential for a nonlinear feedback from η to v. In a turbulent flow case, Waleffe
et al. [30] investigated the different types of nonlinear terms for the particular case of a direct resonance between v and η. They hypothesized that the dominant regeneration process is a result of the nonlinear self-interaction
of the v-terms and that the nonlinear η-terms are small. However, the direct resonance involves higher modes of larger damping which suggests that only a small growth of η is possible compared to the amplifications
obtained from the general transient growth mechanism. For example, in the case of disturbances on a laminar mean flow, Gustavsson [16] reported an induced energy density amplification of 177.9 times for R = 1000
owing to the general transient mechanism. The corresponding value for a direct resonance was 16.5 times.
In pipe Poiseuille flow, the form of 13 implies that it is sufficient to exploit the ansatz e
imαx+θ
to obtain nonlinear -terms in the 8-equations and it is not necessary to use the more general ansatz
P
m
P
n
e
imαx+nθ
. However, just the presence of nonlinear -terms does not automatically imply that the feedback is of
importance. The subsequent analysis has to reveal whether the feedback in 13 effects the development, i.e. does it counteract or support the establishment of a sustained amplification in pipe Poiseuille flow? Naturally,
the model 13 is a very simplified model where for example no generation of higher components is included. It cannot be expected that a very complex phenomenon like laminar-turbulent transition can be fully described
by a model like 13. However, in spite of the simplicity, if the full basic equations governing the stability of pipe Poiseuille flow allow a sustained disturbance amplification, the transition may also be indicated by a
low-dimensional model derived from the basic equations.
The system 13 is solved numerically by a 4th and 5th order Runge–Kutta–Fehlberg integration method. The resulting disturbance development is characterized by the energy density of the disturbances defined by
E
m
= π
Z
1
8
′ m
8
∗
′
m
k
2 m
r
2
+ 8
m
8
∗ m
r
2
+ R
2
m
2
k
2 m
r
2
m
∗ m
r dr 14
normalized by the initial energy density denoted E
m0
. The initial disturbance is defined by giving a
1r
and a
1i
equal values all the others are zero initially and the parameter ε = a
1r
= a
1i
is used to define the initial disturbance strength.
3. Results
3.1. The time development of the disturbances In figure 1 the development of E
1
E
10
is presented for R = 4000, α = 0.03 and ε = 0.004 dashed dotted, ε = 0.0079 dashed and ε = 0.008 solid. For ε = 0.004, the development is as in the linear case, i.e. the
disturbance is transiently amplified to a peak and then decays. The peak occurs at t = 198 and the maximum amplification is 1075. In spite of the few modes included in the model 13, these values coincide well with
the values found in linear studies of α = 0 disturbances where the peak position occurs at tR ≃ 0.049 and the optimal amplification for R = 1000 is E
1
E
10
≃ 72. With R = 4000, the corresponding values are t = 196
and E
1
E
10
= 1152 since the total energy density amplification roughly scales with R
2
for small α. In the
EUROPEAN JOURNAL OF MECHANICS – BFLUIDS, VOL.
18
, N
◦
4, 1999
Self-sustained amplification of disturbances in pipe Poiseuille flow 641
Figure 1. Development of E
1
E
10
for R = 4000, α = 0.03, ε = 0.004 dashed dotted, ε = 0.0079 dashed and ε = 0.008 solid.
case ε = 0.0079, the disturbance does not start to decay immediately after the linear peak is reached as in the previous case. Here, nonlinearity has an influence and the amplification is maintained for a substantially
longer period until the decay rate finally increases and the disturbance rapidly decays. With ε = 0.008, the disturbance behaves at first as in the previous case until t ≃ 700. Then it becomes further amplified to about
2.3 times the linear maximum amplification and a sustained state of amplification continuing for all times becomes established. Although the model 13 solely describes the nonlinear development of disturbances
highly elongated in the streamwise direction, the results indicate that the linear transient growth combined with nonlinearity is capable of causing a sustained amplification of disturbances in pipe Poiseuille flow. For
the same values of parameters as in the sustained case presented in figure 1, the phase-plane development of b
1r
versus b
1i
from 13 is presented in figure 2. The calculations are performed for 0 6 t 6 100 · 10
3
and the star indicates the initial point. Initially, owing to the choice of initial disturbance, b
1r
and b
1i
are zero. After a while, the trajectory follows a fixed track and the sustained state of amplification is represented by
a limit cycle of circular shape. The same behaviour is obtained for other pairs of 13. As a consequence, since for example E
1
E
10
depends on a
2 1r
+ a
2 1i
and b
2 1r
+ b
2 1i
, the energy density becomes constant although the individual components of 13 oscillate. Although it is not formally proven that the sustained amplification
continues for all times, the result showing a sustained limit cycle for t up to 100 · 10
3
makes this plausible. In figure 1, for all three curves, the initial growing phase up to the linear peak is almost identical. However,
the amount of amplification at the linear peak decreases somewhat as ε decreases. The decrease of the transient linear amplification by nonlinearity has also been reported by O’Sullivan and Breuer [31] and by
Bergström [27]. For m = 2, a similiar development as presented for m = 1 is obtained. The energy density of the m = 2 component however is zero initially and the initial growth is induced by the nonlinear interaction of
the m = 1 components.
EUROPEAN JOURNAL OF MECHANICS – BFLUIDS, VOL.
18
, N
◦
4, 1999
642 L. Bergström
Figure 2. Phase-plane development of b
1r
versus b
1i
for ε = 0.008, α = 0.03, R = 4000 and 0 6 t 6 100 · 10
3
.
3.2. Threshold amplitudes of ε for a sustained amplification From numerical tests, the threshold i.e. the lowest value of ε for a sustained amplification versus the
corresponding R is presented in figure 3a for α = 0.03 cross, α = 0.05 star and α = 0.1 circle. The curves are displaced towards larger R as α decreases and the threshold amplitude for sustained amplification begins to
be relatively high for values of R below some thousands. In addition, for α ≪ 1 and lower R but still αR ≫ 1, for example, α = 0.03 and R = 1000, it can be pointed out that no sustained state has been obtained. For
larger α and lower R, sustained amplification can be achieved but the requirement that α ≪ 1 in the derivation of the model becomes violated. In figure 3b the threshold amplitude of ε versus the corresponding value of R is
presented in a logarithmic diagram for α = 0.03 solid, α = 0.05 dashed and α = 0.1 dashed dotted. With an assumed threshold relation of the form ε = constantR
g
, all curves in figure 3b indicate unambiguously that g = −3. In the next section, the indicated value of g will be explained by a heuristic consideration of Eq. 13.
It can be mentioned that in some tests with ε well beyond the lowest ε for sustained amplification, the solution may decay for a narrow interval of ε and then again become sustained outside the interval. This observation
has been made close to the lowest R for which sustained solutions have been achieved for a certain α.
4. Discussion and conclusion
