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УДК 539.374
Analysis of the Dispersion of Hydroacoustic Waves on the Basis of Viscoelastic Model
Vladimir M. Sadovskii* Oxana V. Sadovskaya^ Kristina S. Svobodina*
Institute of Computational Modeling SB RAS, Akademgorodok, 50/44, Krasnoyarsk, 660036
Russia
Received 24.01.2013, received in revised form 24.02.2013, accepted 30.03.2013 On the basis of mathematical model of the Poynting-Thomson viscoelastic medium the effect of acoustic dispersion of water is described: the phase velocity of waves of terahertz frequency is doubled in comparison with the velocity of waves of sound range. Rheological parameters of the model are selected by means of the values of the velocities of propagation of slow and fast monochromatic waves. The system of equations of the dynamics of the Poynting-Thomson viscoelastic medium is reduced to the form, hyperbolic by Friedrichs. It guarantees the correctness of the Cauchy problem and boundary value problems with dissipative boundary conditions, and also allows to use the monotone grid-characteristic schemes for numerical solution of problems. In the framework of 1D model the computations of a transformation of hydroacoustic waves, generated by U-shaped impulse of pressure, were performed. Results of computations show a strong damping of the fast precursor as it passes the distance of hundred nanometers from the moment of entry and the emergence of stable profile of the slow wave at the mesolevel.
Keywords: viscoelastic medium, Poynting-Thomson model, hydroacoustic wave, dispersion, grid-characteristic method.
During the propagation of sound energy an anomalous phenomenon appears in an aqueous medium, discovered by Italian physicists-experimenters nearly twenty years ago. Under normal conditions, the velocity of sound in water is about 1500 m/s and does not depend on its frequency. However, ultrasonic oscillations with a frequency of several terahertz propagate in water at the velocity approximately twice more [1,2]. Recently the physical properties of supercooled water are investigated [3-5]. The fact, that anomalous increase of the velocity of waves in supercooled water occurs at lower frequencies, can be used in elaborating devices and systems of hydroacoustics for the conditions of Arctic Ocean. Generally accepted explanation for this phenomenon does not exist, but there is a hypothesis, according to which such water exhibits viscoelastic properties under the significant increase of frequency and "in the limit" the sound propagates in water with the velocity approximately 3000 m/s, characteristic for ice. Such works as [6,7], etc. are devoted to mathematical modeling of the dispersion of hydroacoustic waves by means of the molecular dynamics method. In present paper the rheological method is applied for the description of this phenomenon.
The rheological scheme of a simple viscoelastic model is represented in Fig. 1. This scheme can not give exhaustive physical explanation of the considered phenomenon, but in the case of correct choice of parameters, included in the model, it is appropriate for technical analysis of the acoustic motions of water, taking into account the wave dispersion. First time such a model was used by Poynting and Thomson for the description of high-temperature deformation of glass
* sadov@ icm.krasn. ru
© Siberian Federal University. All rights reserved
fibers [8]. At high-frequency perturbations a viscous damper, included in the scheme, blocks an elastic spring, which is connected in parallel with it. Therefore the mechanical system is more rigid than at low frequencies, when both springs are deformed. The higher velocity of the propagation of perturbations corresponds to the more rigid system, thus, on a qualitative level, this model correctly describes the dispersion properties of water.
The Poynting-Thomson constitutive equation in a state of uniaxial tension-compression, constructed according to rheological scheme, is formulated in terms of the stress a, the strain e, the rate of change of stress at and the rate of deformation et:
1 + Eo
, n
a + Eat
E o
Ee + net,
(1)
Fig. 1. Rheological scheme of a model
where E0 and E are the Young moduli of elastic springs, included in the scheme, n is the viscosity coefficient of a damper. The subscripts t and x, anywhere they meet, denote the derivatives with respect to time and spatial variable. To simplify the notation, let us consider the Hohenemzer-Prager constitutive equation, which is a more general and no less known in the hereditary elasticity theory:
ao a + a 1 at = bo e + bi et.
(2)
It should be noted that if the coefficients are calculated according to the equation (1), then the inequality A = a0 b1 — a1 b0 > 0 holds automatically. This inequality is the condition of positivity of the internal energy dissipation [9]. Moreover, if all coefficients of the equation (2) are of the same sign and satisfy this condition, then considered models coincide exactly. Indeed, the constitutive equations (1) and (2) are identical, when the system
Eo + E
n
E
bo
ao Eo a1 Eo
is fulfilled, the unique solution of which has the next form:
Eo = —, a1
E=
bo bi A
bi
n=
bi A.
Distinction of constitutive equations takes place only if signs of the coefficients ao, a1, bo and b1 are not the same. In this case the Hohenemzer-Prager model does not have correct rheological scheme.
Complete system of equations for the description of one-dimensional motions with plane waves is obtained by adding to (2) the equation of motion and the kinematic equation:
pvt = ax
et vx
(3)
where p is the density of a medium. Double differentiation of (1) with respect to time allows to reduce this system to the third-order equation for the stress:
bo b1
ao att + a1 attt — axx + axxt.
pp
Substituting the expressions for a monochromatic wave
a = a e*M-fcx) = a efcixei(wt-feox) = a efci xei^(t-x/c)
(w is the circular frequency, k = ko + ik1 is the complex wave number, c = w/ko is the phase velocity), we obtain the dispersion equation:
pao(i w)2 + pa1(iw)3 = bo(—ik)2 + i w b 1 (—ik)2.
From this equation one can find the formulas for calculation of the wave number, the wave velocity and the wave damping decrement as the wave propagates in positive direction of the axis x:
ao + i w a1 bo + i w b1
a/P Re
ao + i w a1 bo + i w b1
A = --
1
Im k
Consequently, under low frequencies w ^ 0 ^ c ^ \Jbo/(ao p) = co, under high frequencies w ^ to ^ c ^ \Jb1/(a1 p) = cO. Two of three parameters of the model can be determined by the values co and cO of velocities of slow and fast waves. In order to find the third parameter, it is necessary to calculate the damping decrement of high-frequency waves — a characteristic distance, with the passage of which the amplitude decreases by e times: AO = — 1/kf°, where
,0 r T i 1 i- t V1 — i ao/(wa1) 1 y T 1 — iao/(2 wa1)
ko = lim Im k = — lim w Im _____________ ___= — lim w Im-----------------------
w—O cw U—O ^/1 — ibo/(wb1) cTO u—° 1 — ibo/(2wb1)
1 (1 — iao/(2 wa1^(1 + i bo/(2 wb1))
— lim w Im---------------
A
1 + b2/(4 w2b2)
2 a1b1 cO
The parameter AO can be chosen taking into account the fact that the transition to fast sound in water occurs under a frequency of the megahertz range. The formulas for recalculation of phenomenological parameters take the following form:
Eo
pcO,
E
1
p c2o
1
PcOo
-1
n
AO
~2
pc0
The analysis shows that an adequate description of the phenomenon can be achieved only by setting the wave damping decrement of the order of 10 nanometers. In Figs. 2 and 3 one can see the dispersion dependencies for the different frequency ranges, from sonic and ultrasonic to hypersonic. Here the frequency v = w/2 n.
Fig. 2. Dependence of the wave velocity c on the frequency v for ultrasonic range
1
c
c
OO
Fig. 3. Dependence of the wave velocity c on the frequency v for hypersonic range
The series of curves correspond to the values of ATO between zero and 10 nanometers with the step of one nanometer. If a significantly greater value of the damping decrement is given, then the transition to the "fast sound" will occur at very low frequencies, that is not consistent with the experimental observations. Note, that an estimate of the decrement from physical considerations, worked out in [5], gives rather small values, too.
The system of equations (2), (3) can be written in the matrix form:
p 0 0 \ d 1 V
0 1 0 £
0 0 ai 1di \ ^ a
0 0 1 1
d
00 dX bi 0 0 '
Multiplying on the left by the nonsingular matrix
T =
A
0
0
bo bi
1
A
0 0 1 000 100
A p 0 0
0 bo bi —aibo
0 —aibo ao ai
- ( dx I a)— A (0 A 1 0
V
£
a
+
0 0 V
0 0 £
bo —ao a
0
bo
0 —aibo ao
one can reduce this system to the symmetric form:
d
di
V
£
a
b0 —ao bo
— a0 bo ao
(5)
It is easy to verify that the dissipativity condition A > 0 of the model provides a positive definiteness of matrix under the time derivatives. Thus, the system of equations (5) belongs to a class of systems, hyperbolic by Friedrichs [10,11]. For these systems the Cauchy problem and the boundary-value problems with dissipative boundary conditions are well-posed. Under numerical solution of problems it is possible to apply the well-developed shock-capturing algorithms, based on modifications of the Godunov discontinuity decay method. It is essential that these algorithms have the property of monotonicity, which is characterized by the absence of spurious oscillations of the solutions at fronts of discontinuities and in narrow zones of high gradients.
0
0
Computational results for the problem about one-dimensional motion of water with plane waves were obtained on the basis of the grid-characteristic method with the maximum allowable time step by the Courant-Friedrichs-Lewy condition. The method of two-cyclic splitting with respect to physical processes is used, which includes three stages. At the first and third stages, the system of equations (5), in which the terms not containing derivatives are omitted, is solved by means of the Godunov method. At the second stage, the system of ordinary differential equations, obtained from (5) after deleting the derivatives with respect to spatial variable, is solved by means of the Crank-Nicholson scheme.
The results of computations of the problem about the action of a U-shaped impulse of stress on the boundary of a nanoscale layer for three different time moments are represented in Fig. 4. The thickness of a layer is equal to 10 nm, the impulse duration is 0.2 ■ 10-9 s. The upper curves characterize the normal stress a, and the lower ones characterize the velocity v of particles of a medium. The spatial variable in the direction of horizontal axis is dimensionless with respect to the layer thickness. Parameters of the model are defined according to the values of velocities of propagation of fast and slow sound in water. The damping decrement is equal to 10 nm. The graphs show, how the fast precursor of acoustic wave damps, as it passes deeper into the layer.
-l
-2
a
V X
-1
-2
a
——
V X
-1
~2[
a
V ! x
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 O.i
Fig. 4. Damping of the amplitude of the fast precursor (l = 10 nm)
The next series of graphs in Fig. 5 is the solution for a layer of the thickness 100 nm. Under this scale the curves, smoothly decaying almost to zero, correspond to the damped precursor. The slow wave arises after the precursor, it moves with the velocity two times less than the velocity of precursor.
1.5
0.5
0
-0.5
-1.5
-2.5
a
/
V X
1.5
0.5
0
-0.5
-1.5
-2.5
a
V X
1.5
0.5
0
-0.5
-1.5
-2.5
a
V x
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 O.i
Fig. 5. Emergence of the slow wave (l = 100 nm)
The solution for a layer of the thickness 1 ^m is shown in Fig. 6. At the mesoscale level the slow wave takes a U-shaped form, which can be explicitly derived from the equations of acoustics, not taking into account the dispersion of waves.
Thus, the general picture of wave motion of a medium under the action of impulse of stress consists in the appearance of a fast damped precursor, which turns into an elastic wave, propagating in water at the velocity 1500 m/s.
1.5
0.5
0
-0.5
-1.5
-2.5
a
V X
0 0.2 0.4 0.6 0.8
1.5
0.5
0
-0.5
-1.5
-2.5
0 0.2 0.4 0.6 0.8
1.5
0.5
0
-0.5
-1.5
-2.5
a
V X
0 0.2 0.4 0.6 0.8
Fig. 6. Propagation of the slow wave (l = 1 ^m)
Finally, we present three-dimensional variant of the model with constitutive equations of the Poynting-Thomson viscoelastic medium, on the basis of which it is possible to describe a phenomenon of dispersion of the spatial hydroacoustic waves of small amplitude:
pvt = V a, £t = V ■ v, ao a + ai at = bo e + bi £t,
where v is the vector of velocity, a is the hydrostatic stress, e is the volume strain, V is the Hamilton operator, the dot between vectors is used to denote the scalar product of vectors.
This work was supported by the Russian Foundation for Basic Research (grant 11-01-00053) and the Complex Fundamental Research Program no. 18 of the Presidium of RAS.
References
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[3] O.Mishima, H.E.Stanley, The relationship between liquid, supercooled and glassy water, Nature, 396(1998), 329-335.
[4] P.G.Debenedetti, H.E.Stanley, Supercooled and glassy water, Physics Today, 56(2003), no. 6, 40-46.
[5] P.A.Starodubtsev, R.N.Alifanov, S.V.Gutorova, Modern methods for transmission of sound energy in water and their theoretical development, Journal of Siberian Federal University. Mathematics and Physics, 2(2009), no. 2, 230-237 (in Russian).
[6] A.Rahman, F.H.Stillinger, Propagation of sound in water. A molecular-dynamics study, Phys. Rev. A, 10(1974), no. 1, 368-378.
[7] M.P.Allen, D.J.Tildesley, Computer Simulation of Liquids, Clarendon Press, Oxford, 1987.
[8] M.Reiner, Deformation, Strain and Flow: an Elementary Introduction to Rheology, ed. H.K.Lewis, London, 1960.
[9] O.Sadovskaya, V.Sadovskii, Mathematical Modeling in Mechanics of Granular Materials, Ser.: Advanced Structured Materials, Vol. 21, Springer, Heidelberg-New York-Dordrecht-London, 2012.
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Анализ дисперсии гидроакустических волн на основе модели вязкоупругой среды
Владимир М. Садовский Оксана В. Садовская Кристина С. Свободина
На основе математической модели вязкоупругой среды Пойнтинга-Томсона описан наблюдаемый в физических экспериментах эффект акустической дисперсии воды, суть которого заключается в удвоении фазовой скорости волн терагерцовой частоты по сравнению со скоростью волн звукового диапазона. По значениям скоростей распространения медленных и быстрых монохроматических волн подобраны реологические параметры модели. Система уравнений динамики вязкоупругой среды Поинтинга-Томсона приведена к гиперболической по Фридрихсу форме, что является гарантией корректности постановки задачи Коши и краевых задач с диссипативными граничными условиями, а также дает возможность применения монотонных сеточно-характеристических схем сквозного счета при численном решении задач. В одномерной постановке выполнены расчеты трансформации гидроакустической волны, генерируемой П-образным импульсом давления на разных масштабных уровнях, результаты которых демонстрируют сильное затухание быстрого предвестника по мере прохождения им расстояния в сотню нанометров с момента вступления и зарождение устойчивого профиля медленной волны на мезоуровне.
Ключевые слова: вязкоупругая среда, модель Пойнтинга-Томсона, гидроакустическая волна, дисперсия, сеточно-характеристический метод.
