da
dt
= a μ R − R c
ð
Þ− ν a
j j
2
where R is a control parameter, R c is its critical value, ν is a parameter of nonlinear
decay. Amplitude of steady state oscillations «a» satisfies the following equation:
R − R c
ð
Þ−
ν
μ
a
j j
2
a = 0.
Experimentally measured dependence of parameter R proves the excitation of a
globally unstable mode on a jet (Fig. 4) [21].
The analysis shows that IW can be generated as a result of self-sustained
oscillations of a plume in the pycnocline. Self-sustained oscillations develop in a
system characterized by absolute instability. It is well-known that free shear flows
are convectively unstable. It has been shown in [29] that the type of instability can
change to absolute if the hydrodynamic flow is lighter than the surrounding fluid or
due to the presence of a counterflow. Thus, a plume in the pycnocline can be
absolutely unstable due to the counterflow.
Linear stability analysis of the fountain in the pycnocline was performed in [10,
21]. The earlier analysis was based on the experiment in a small stratified tank using
Particle Image Velocimetry [10]. This experiment was intended to demonstrate the
effect, while the parameters did not provide scale modeling of wastewater outfall.
Assuming the flow in the pycnocline locally quasi-parallel, the authors performed a
theoretical linear stability analysis assuming that the mean velocity profiles in the
subsequent cross-sections present the basis flow (Fig. 5).
They have demonstrated that there is a region of absolute instability, while
self-sustained oscillations can develop in the plume. Moreover, theoretically estimated frequencies of an absolutely unstable axisymmetric mode showed good
correspondence with experimentally measured frequency of plume oscillations and
Fig. 4 Isotherm
displacement variance in the
thermocline versus discharge
rate: a in the upper part of the
thermocline (isotherms 13–
19 °C) and b in the lower part
of the thermocline (isotherms
9–12 °C). Notations 1 and 2
correspond to two series of
experiments
72
V. G. Bondur et al.
dt
= a μ R − R c
ð
Þ− ν a
j j
2
where R is a control parameter, R c is its critical value, ν is a parameter of nonlinear
decay. Amplitude of steady state oscillations «a» satisfies the following equation:
R − R c
ð
Þ−
ν
μ
a
j j
2
a = 0.
Experimentally measured dependence of parameter R proves the excitation of a
globally unstable mode on a jet (Fig. 4) [21].
The analysis shows that IW can be generated as a result of self-sustained
oscillations of a plume in the pycnocline. Self-sustained oscillations develop in a
system characterized by absolute instability. It is well-known that free shear flows
are convectively unstable. It has been shown in [29] that the type of instability can
change to absolute if the hydrodynamic flow is lighter than the surrounding fluid or
due to the presence of a counterflow. Thus, a plume in the pycnocline can be
absolutely unstable due to the counterflow.
Linear stability analysis of the fountain in the pycnocline was performed in [10,
21]. The earlier analysis was based on the experiment in a small stratified tank using
Particle Image Velocimetry [10]. This experiment was intended to demonstrate the
effect, while the parameters did not provide scale modeling of wastewater outfall.
Assuming the flow in the pycnocline locally quasi-parallel, the authors performed a
theoretical linear stability analysis assuming that the mean velocity profiles in the
subsequent cross-sections present the basis flow (Fig. 5).
They have demonstrated that there is a region of absolute instability, while
self-sustained oscillations can develop in the plume. Moreover, theoretically estimated frequencies of an absolutely unstable axisymmetric mode showed good
correspondence with experimentally measured frequency of plume oscillations and
Fig. 4 Isotherm
displacement variance in the
thermocline versus discharge
rate: a in the upper part of the
thermocline (isotherms 13–
19 °C) and b in the lower part
of the thermocline (isotherms
9–12 °C). Notations 1 and 2
correspond to two series of
experiments
72
V. G. Bondur et al.