Full text of DOI:10.1134/1.1333872

Doklady Physics, Vol. 45, No. 11, 2000, pp. 627–631. Translated from Doklady Akademii Nauk, Vol. 375, No. 3, 2000, pp. 338–342.
Original Russian Text Copyright © 2000 by Chashechkin, Kistovich, Il’inykh.
MECHANICS
Experimental Study of the Generation of Periodic Internal Waves
by the Boundary Layer on a Rotating Disk
Yu. D. Chashechkin, Yu. V. Kistovich, and Yu. S. Il’inykh
Presented by Academician A. Yu. Ishlinskiœ February 10, 2000
Received February 11, 2000
The exact solution to the linearized problem of the was determined by the density tag method [5] and was
generation of internal waves, which involves internal found in our experiment to be Tb = 7.5 s. Observations
waves and internal boundary flows [1], allows us to
estimate errors intrinsic to the well-known method of
force (or mass) sources [2]. In the case of small dis-
placements, calculations of perturbances excited by an
oscillating bar satisfactorily agree with the measure-
ment results of [3]. There exist situations when a body
moving periodically in continuously stratified viscous
fluid does not radiate (in the linear case) and generates
only isopycnic boundary flows. This situation takes
place, e.g., in the case of a horizontal disk performing
torsional vibrations [4]. However, by virtue of the non-
linear nature of the hydrodynamic system of equations,
various forms of motion interfere with one another. In
particular, thin-layer boundary flows can be a source of
periodic waves [4]. Previously, experimental studies of
such internal-wave generators were not conducted.
Therefore, it is of interest to investigate the practical
feasibility of the principles for nonlinear generation. In
the present paper, the possibility of generation of three-
dimensional beams of periodic internal waves by tor-
sional vibrations of a horizontal disk are studied and the
principal regularities connecting wave-field character-
istics with the properties of a medium and source
motion parameters are established.
of the flow pattern in the vertical plane were performed
by the shadow IAB-451 device using the method called
“the vertical slit-thread in focus” [6]. Due to the sym-
metry of the flow pattern in the linearly stratified
medium, the shadow device visualizes the perturbance
distribution in the central vertical axis, where the light
beam passes along the tangent to the wave phase sur-
faces. The rest contributions initiated by perturbances
along the beam are mutually compensated. The mea-
surements of wave displacements were carried out by
an electrical-conduction single-electrode sensor and
using sweep-vibration methods [7]. The sensor was cal-
ibrated before each experiment in accordance with the
lifting-submersing procedure. The error of wave-dis-
placement measurement did not exceed 20%.
The wave source was a horizontal disk 1 mm thick
and 2 (or 4) cm in diameter. The disk was fixed to a ver-
tical rod 2 mm in diameter, which was connected
through a reducer to a dc motor. To reduce perturbances
of the medium, the rod was placed in an immobile tube
6 mm in diameter. Adjustment of the rotation frequency
and the law of disk motion was performed by varying
the voltage applied to the motor. The angular displace-
ment of the motor was recorded by a multiturn potenti-
ometer. Three types of disk motion were studied: tor-
sional harmonic vibrations (with the angular velocity
Ω = Ω0sinω*t); intermittent alternate rotations (meander)
The experiments were performed in a laboratory
wave channel with dimensions 9.0 × 0.6 × 0.6 m filled
with an exponentially stratified solution of common
salt, which had transparent windows made of optical
glass. The period (for the frequency N) of the buoyancy
T
+Ω0, nT < t < nT + ---
2
2π
N
Λ
g
Ω = Ω0
Tb = ------ = 2 π --- ,
T
2
– Ω 0 , nT + --- < t < nT + T,
–1
dlnρ0
-------------
dz
where Λ =
is the stratification scale, ρ0(z) is
where T = 2π/ω* ; and harmonic torsional vibrations
against the background of a uniform rotation (Ω = A +
Ω0sinω*t). The maximum linear velocity for the disk
edge motion was U = 10 cm/s.
the density profile, and g is the free-fall acceleration,
The flow shadow pattern appearing after the com-
pletion of two 4-cm-disk vibrations is shown in Fig. 1.
Among the optical inhomogeneities of the pattern, we
Institute of Problems in Mechanics,
Russian Academy of Sciences,
pr. Vernadskogo 101, Moscow, 117526 Russia
1028-3358/00/4511-0627$20.00 © 2000 MAIK “Nauka/Interperiodica”

628
CHASHECHKIN et al.
Fig. 1. Shadow pattern of a flow formed by a horizontal disk performing torsional vibrations; T = 7.5 s, T = 20 s, U = 2 cm/s.
b
can distinguish a sequence of thin strips positioned in flow separates from the disk and begins to attenuate. As
parallel to the disk, which are linked by three embedded far as the ring-shaped flow is being damped, its vertical
ring-shaped structures. Two systems of inclined, almost dimension decreases and its vortices collapse. In the
horizontal, diffusive dark and light strips branch out of case of changing the rotation direction (at every half of
the external surface of the layered structure. In the a period), a new pair of boundary flows and, corre-
upper half space, the interior strip is dark, whereas in spondingly, a pair of new monotonically raising annu-
the lower one, the opposite situation takes place. Hori- lar flows arise. In each of these flows, the fluid takes
zontal strips are antisymmetric to both the right and the part in two types of rotary movement, namely, around
left. The positions of the horizontal interlayers, as well the vertical disk axis in the horizontal plane and in the
as the size and shape of the vortices linking these inter- vertical plane around the circular symmetry axis. As in
layers, and the mutual positions of the dark and light the uniform medium, the disk forms the middle flow
inclined strips vary with time.
[9], in which the fluid flows along the vertical axis and
is thrown away in the disk plane forming a wave beam
of zero frequency in the stratified medium. The direc-
tion of the middle flow induced by the disk is indepen-
dent of its rotation direction.
Comparison with the results of [3, 8] for shadow
observations of stratified flows shows that inclined dif-
fusive strips represent schematically periodic internal
waves with frequency ω = 2ω* = 0.62 s–1, which prop-
agate at an angle θ = 47° to the horizon, as well as dif-
fusive horizontal strips, i.e., waves of the zero fre-
quency, and high-gradient interlayers, i.e., shells of
rotating ring-shaped flows at the disk edge. The distri-
bution of optical perturbances in the shadow image,
when the blackening density is determined by the hori-
zontal component of the refractive index (or the den-
sity) bound by the linear relationship [8], testifies to the
existence of an antisymmetric pattern of periodic
waves: a crest in the upper half space corresponds to a
hollow in the lower one and vice versa.
In the shadow pattern (Fig. 1), we observe traces of
three pairs of annular flows. The first (outer) flow pair
has collapsed under the action of buoyancy forces; the
second (central) pair has a clearly expressed annular
structure similar to the vortex structure in a homoge-
neous fluid [10]; and the third pair intervenes in them.
The external boundaries of annular flows are seen
above and below the disk as thin horizontal interlayers.
Their thickness does not exceed 0.3 mm and, in fact, is
determined by the resolution of the shadow device [6].
The shape of the exterior annular flow demonstrates
that the mixing of the fluid inside this flow is weak,
since its fragments come to the disk horizon within the
observation time.
Observations of the flow pattern show that at the ini-
tial phase of the flow, boundary flows arise at both disk
sides; each of these flows forms a circular flow rotating
together with the disk, the cross section of this flow
The outer boundary of annular flows being formed
having an annular shape. The large and small ring radii near the disk radiates two groups of axisymmetric peri-
monotonically increase with time up to the moment of odic internal waves at a frequency ω = 2ω*. These
changing the rotation direction. At this moment, the waves propagate at an angle θ = 47° to the horizon. Fur-
DOKLADY PHYSICS Vol. 45
No. 11
2000

EXPERIMENTAL STUDY OF THE GENERATION OF PERIODIC INTERNAL WAVES
629
η, mm
0.1
0
1
2
–0.1
3
4
U, V
5
0
–5
η, mm
0.1
0
2
1
–0.1
U, V
5
3
5
4
0
–5
0
200
400
T, s
Fig. 2. Recorded signals of electrical-conductance and disk position sensors.
thermore, all elements of this pattern are repeated in sional-vibration amplitude [(1–4): η = 0.01, 0/03, 0/06.
0.11 cm].
every half of a period and the common-rotation direc-
tion of annular vortices in the horizontal plane changes.
The lower pair of the records in Fig. 2 corresponds
to the case when, in addition to the alternating voltage,
the permanent voltage is applied to the motor. The
value representing this voltage consequently becomes
equal to the harmonic-signal amplitude and, further-
more, exceeds it. In this case, torsional harmonic vibra-
tions of the 2-cm disk (Usinω*t, U = 2.8 cm/s, ω* =
0.31 s–1) are added to the unidirectional rotation (with
velocity values 1–5, respectively, of 0, 1.0, 3.5, 6.0, and
9.8 cm/s) and are gradually transformed into motion
with the alternate angular velocity of the constant sign
(U0 > U). In the sensor signal which is initially har-
monic and has the frequency 2ω*, a noticeable compo-
nent appears that becomes dominating with further
increase in the permanent rotation component (see seg-
ment 3). In this case, the energy of emitted waves also
increases.
The signals registered from the electrical-conduc-
tance sensor installed in the internal-wave beam at a
distance of 8 cm from the disk (the sensitive element of
the sensor is depicted in the upper right angle of Fig. 1)
are shown in Fig. 2. In the same figure, the signals from
the disk angular position for which a multiturn potenti-
ometer is used is also presented. The upper pair of
records corresponds to the case when the 4-cm disk
performs harmonic torsional vibrations at a constant
frequency and with different amplitudes [(1–4): U =
0.4, 1.3, 3.5, and 5.8 cm/s]. Stable internal waves are
excited at all velocities of disk motion. Comparing the
upper pair of records, we can see that the frequency of
the radiated wave exceeds that for the torsional vibra-
tions by a factor of two. The small amplitude modula-
tion of the signal is initiated by seiches existing in the
basin. The drift of the potentiometer signal with time is
caused by the specific asymmetry of the control signal
and by features of the motor operation in itself. The
The dependence of the vertical displacements in the
internal-wave beam on the maximum velocity of the
disk edge in the case of purely torsional vibrations is
internal-wave amplitude increases linearly with the tor- presented in Fig. 3 (the sensor is placed at a distance of
DOKLADY PHYSICS Vol. 45 No. 11
2000

630
CHASHECHKIN et al.
the order of 0.05 mm) caused by the wave background
η, mm
of the basin. This background is formed due to the
action of other independent sources (mechanical vibra-
tions of the ground, as well as vibrations of the base and
walls of the basin, which are caused by the action of the
wave-producing device). The level of residual vibra-
tions in Fig. 4 corresponds to the modulation depth in
Fig. 2.
1
0.4
As was shown in the experiments, the amplitude of
the wave being radiated substantially depends on the
characteristics of disk motion (under the condition of
preserving the frequency and amplitude for its periodic
component). When the envelope is a meander, only
annular flows and waves are observed in the flow pat-
tern (running internal waves are absent).
As is shown in the theory of the excitation of inter-
nal waves by the horizontal disk performing the tor-
sional vibrations in a stratified viscous fluid [4], the true
wave source is an isopycnic boundary flow character-
ized by a single azimuth velocity component uϕ . The
stream function Ψ for the beam of radiated waves sat-
isfies the equation
0.2
2
0
2
4
6
8
U, cm/s
Fig. 3. Vibration amplitudes for particles on the internal-
wave beam axis as a function of the disk edge rotation veloc-
ity: (1) D = 4 and (2) 2 cm.
η, mm
2
2
2
∂
∂
1
∂
1
2
------
----
∂t
∆ – --- + N ∆ – ------- – ν
∆ – ---
Ψ
∂t2
r2
∂z2
r2
(1)
0.1
∂2u2
r ∂t∂z
1
ϕ
------------
=
,
where ∆ is Laplacian, ν is the kinematic viscosity, and
the origin of the cylindrical coordinate system (r, ϕ, z)
is taken in the disk center. The solution to this equation
with exact boundary conditions describes both running
waves [4] and zero-frequency waves. The form of the
right-hand side of Eq. (1) implies that the frequency of
waves excited by harmonic torsional vibrations of the
disk is twice that of their own frequency. The meander
squared is a quantity independent of time, which
explains the radiative inefficiency of the periodic
motion of this type.
0.04
3
5
10
q, cm
Fig. 4. Vibration amplitude for particles on the beam axis as
a function of the distance to the radiator (T = 7.5 s,
D = 4 cm, T = 16.7 s, U = 2 cm/s).
b
8 cm from the disk along the beam axis). At low rota-
tion velocities of the disk (U < 3 cm/s), the wave ampli-
tude increases quadratically for the radiators of both
diameters (D = 2 and 4 cm: η = 0.005U2 and 0.04U2; η
and U are expressed in millimeters and centimeters per
second, respectively, see curves 1 and 2). At intermedi-
ate velocities, the amplitude-increase rate is gradually
slowed, and for U > 8 cm /s, saturation takes place
owing to the combined action of the nonlinearity and
viscosity effects.
It follows from the solution to Eq. (1) that torsional
vibrations of a disk whose radius is smaller than the vis-
3
cous wave scale LV = gν/N excite a single-mode
wave beam. The displacements of the particles along
the beam axis are
1/6
U2R2 sinθ
2cos4 θ
7
--
6
--------------------------- ------------------------
h(q) =
Γ
,
(2)
48(1 + 2) π3ν4N8q10
The plot illustrating the attenuation of particle dis-
placement amplitudes at the beam axis as a function of
the distance to the radiator is presented in the double-
logarithmic scale in Fig. 4. Making use of the least-
squares method shows that for q < 7 cm, experimental
where Γ is the gamma function and q is the distance
along the beam axis to the source. The quadratic depen-
dence of the wave amplitude on the velocity U in (2) is
consistent with the measurement data exhibited in
Fig. 3. In this theory, the wave amplitude also depends
points are grouped around the curve η = 1.14q–1.54. At quadratically on the disk radius. The ratio of the coeffi-
longer distances (q > 7 cm), the recorded wave ampli- cients in formulas interpolating experimental data of
tude attains saturation (the vertical displacement is on Fig. 3 is equal to 8, whereas the ratio squared for disk
DOKLADY PHYSICS Vol. 45 No. 11
2000

EXPERIMENTAL STUDY OF THE GENERATION OF PERIODIC INTERNAL WAVES
631
diameters is 4. This discrepancy is associated with the the Russian Foundation for Basic Research, project
fact that under the experimental conditions, the waves no. 99-05-64980.
excite both isopycnic boundary flows and annular rotat-
ing vortices. The efficiency of the second generation
REFERENCES
mechanism that is not taken into account in the theory
1. Yu. D. Chashechkin and Yu. V. Kistovich, Dokl. Akad.
elevates with increasing disk diameter. The decrease in
the beam amplitude with the distance from the source,
which is calculated in accordance with (2) under the
experimental conditions presented in Fig. 4, is given by
the expression h = 1.29q–5/3, where h and q are
expressed in millimeters and centimeters, respectively.
For q < 7, empirical data can be approximated by the
dependence h = 1.14q–1.54. Deviations in the indices
(8%) and coefficients (13%) do not exceed the mea-
surement error for the wave amplitudes by the contact
electrical-conductance sensor, which attains 20% in the
experiments under discussion. Thus, the disk perform-
ing high-amplitude torsional vibrations in the continu-
ously stratified fluid efficiently radiates periodic inter-
nal waves.
Nauk 355, 54 (1997) [Phys. Dokl. 42, 377 (1997)].
2. M. G. Lighthill, Wave in Fluids (Cambridge Univ. Press,
Cambridge, 1978; Mir, Moscow, 1981).
3. Yu. S. Il’inykh, Yu. V. Kistovich, and Yu. D. Chashech-
kin, Izv. Akad. Nauk, Fiz. Atm. Okeana 35, 649 (1999).
4. Yu. V. Kistovich and Yu. D. Chashechkin, Dokl. Akad.
Nauk 367, 636 (1999) [Dokl. Phys. 44, 573 (1999)].
5. S. A. Smirnov, Yu. D. Chashechkin, and Yu. S. Il’inykh,
Izmer. Tekh., No. 6, 15 (1998).
6. L. A. Vasil’ev, Schlieren Methods (Nauka, Moscow,
1968; Israel Program for Scientific Translations, Jerusa-
lem, 1971).
7. A. V. Gvozdev, V. I. Neklyudov, andYu. D. Chashechkin,
Izmer. Tekh., No. 3, 33 (1990).
8. D. E. Mowbray, J. Fluid Mech. 27, 595 (1967).
The method described expands the spectrum of possi-
ble means for excitation of internal waves including wave
generation under the action of viscous stresses [11],
as well as by varying the volume and position of a
body [12].
9. H. Schlichting, Theory of Boundary Layers (McGraw-
Hill, New York, 1955; Nauka, Moscow, 1960).
10. Album of Fluid Motion, Ed. by M. VanDyke (Parabolic
Press, Stanford, 1982; Mir, Moscow, 1986).
11. A. V. Kistovich and Yu. D. Chashechkin, Prikl. Mekh.
Tekh. Fiz. 40 (6), 31 (1999).
ACKNOWLEDGMENTS
12. S. A. Makarov, V. I. Neklyudov, andYu. D. Chashechkin,
Izv. Akad. Nauk SSSR, Fiz. Atm. Okeana 26 (7), 744
(1990).
This work was carried out under the financial sup-
port of the Ministry of Science and Technology of the
Russian Federation (in the framework of the Program
of Supporting Unique Experimental Facilities) and of
Translated by G. Merzon
DOKLADY PHYSICS Vol. 45
No. 11
2000