AP02_04.vp 1 Introduction The mean temperature distribution in resonators with high amplitudes of the acoustic field is caused by thermo- dynamic state changes due to the mean pressure and the density changeover, and in addition due to the conversion of acoustic energy into heat due to the viscous losses. The mean temperature distribution needs to be known in some ap- pliances, such as thermoacoustic engines. The one-dimens- ional model equation for nonlinear standing waves of the 2nd order including a viscous boundary layer is modified to describe the acoustic standing waves in a resonator with a longitudinal distribution of mean temperature. It is assumed that the mean temperature changes are small. 2 Model equations Let us consider a one-dimensional acoustic field in an axisymetrically shaped gas-filled resonator driven by means of an external force. The model equation describing the acoustic field is derived from the basic equations of fluid mechanics. The one-dimensional form of these equations with terms up to the 2nd order is presented here: Navier- -Stokes equation, see [1], � � � � � � � � � � � � � � � � 0 0 2 02 4 3 v t v t v x a a p x � � � � � � � � � � �� � � � � � � �x r x r v 1 2 2� �� � �� , (1) the continuity equation taking into account the boundary layer, see [2, 3], � � �� � � � � � � � �� � � � � � � � � � � � � �� � t r x r v r x r v v x r 0 2 2 2 2 3 0 2 1 1 Pr � � � � � � � 0 212 1 2 r v t x , (2) where the fractional derivative represents an integrodifferen- tial operator � � � � � � �� � � � � �� 1 2 1 2 1f t f t t t t t d (3) and the thermodynamic state equation taking into account heat conductivity and mean temperature change, see [4], � � � � � � � � � � � � � � � � � p c c c c c r p V p 2 0 0 2 0 2 1 2 1 1 1 1 � � � � � � �� � 2 2 � �x r v , (4) where � �� , p v, are acoustic density, pressure and velocity, p0 0 0, ,� � are equilibrium state pressure, density and tem- perature, � is mean temperature, �� � � ��, � a a t� is the driving acceleration, x is the spatial coordinate along the reso- nant cavity, t is time, � r r x� is radius of the resonator, � � c cVp is ratio of specific heats at constant pressure and volume, is the coefficient of thermal conduction, �0 is kinematic viscosity, �, � are the coefficients of bulk and shear viscosity, Pr is the Prandtl number, c0 is the small-signal sound speed due to equilibrium temperature �0 and c is the small-signal sound speed due to changed mean temperature defined as c c c2 0 2 0 0 2 0 1� � � � �� �� � �� � � � . After linearization, from equations (1), (2) and (4) we obtain the following relations with terms of the 1st order � � � �� � � �p t ax� �� � 0 , (5) � � �� � � � � �� � � �� � � �0 2 2 0 1 c t ax c c p �� , (6) 1 1 2 2 2 2 r x r x c t x a t � � �� � � � � � � � � � � � � � � � � � d d , (7) where � is velocity potential, v t� �� � . For deriving the model equation, the following method is used: linearized equations, Eqs. (5), (6), (7), are substituted into terms of the 2nd order and so the resulting error is of the 3rd order. It is assumed that the mean temperature changeover is small and that its time and spatial derivatives are of the 2nd order. After eliminating acoustic density �� from Eq. (1) using Eq. (6), introducing velocity potential and after its integra- tion with respect to the spatial coordinate x we obtain the relation 8 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 42 No. 4/2002 Temperature Effects in Acoustic Resonators M. Červenka, M. Bednařík, P. Koníček This paper deals with problems of nonlinear standing waves in axisymetrically shaped acoustic resonators where a mean temperature is distributed along the axis. Keywords: nonlinear standing wave, acoustic resonator, temperaturechangeover. � � �� � � �� � � � �� � � � � t c t ax c c t ax p � �� � � � � � � � � � 2 1 2 2 2 �� � � � � � � � � � � � � � � � � � � � �� � � � � � � � � �2 4 32 2 2 2x ax p c t x a t d d � � . (8) We can eliminate pressure �p from Eq. (8) using Eq. (4). Thus we get � � �� � � �� � � � �� � � � � t c t ax c c t ax p � �� � � � � � � � � � � 2 1 2 2 2 � � � � � � � � � � � � � � � � � �� � � � � � � � � � � �2 1 2 2 2 2 2 x c c ax b c t x p �� � � � d d a t c c t ax cp � � � � � � � � � � � � �� � �� � � �� � �� 0 2 4 2 2 1 1 �� . (9) After differentiating Eq. (9) with respect to time, we eliminate the time derivative �� �� t using Eq. (2), thereby obtaining a model equation describing the nonlinear stand- ing waves in an acoustic resonator with a spatial distribution of mean temperature. The derived model equation has the form � � � � � �� � � � �� � 2 2 0 2 0 2 2 21 t c r x r x t x x� � � � � � � � � � � � � � da t a x c t t ax b d � � � � � � � � � � � � � � � �� � � � � �� � � � 1 2 0 2 0 2 2 � � c t x a t r 0 2 0 3 3 2 0 2 1 1 3 2 � � � � � � � � � � � � � � � � � � �� � � � d d 2 Pr � � � � � t x a t c c p 3 2 0 2 2 0 2 2 1 � � � � � � � � � � � � � � � � d d 1 2 1 2 �� � � � �t x a t2 � � � � � � � d d , (10) where � b c cV� � � �� � 4 3 1 1 p is the diffusity of sound. This model equation is written in the coordinates moving together with the resonator body, consequently the boundary conditions have the form �� �x v� � 0 for x � 0 and x � �, where � is the resonator cavity length. The acoustic pressure and density can be calculated from the numerical solution of Eq. (10), applicable equations are derived from relations (8) and (9) � � � � � � � � � � � � � �p t ax x c t ax� �� � � � �� � � �� � � � � � 2 2 2 0 2 0� � � � � �� � � � � � � � � � � � � 2 0 2 0 2 2 0 1 4 3c t x a t c c p � � � � � � � d d � 2 01� � �� � � � � � � � ��� � � t ax , (11) � � � � � � � � � � � � �� � � � �� � � � � c t c ax c x0 2 0 0 2 0 0 2 0 2 2 � � � � � � � � � c c c c b c p p 0 2 0 2 0 4 3 2 0 3 0 4 0 1 2 1� � � �� �� � � � � � � � � �� � � �� � � � � � � � �� � � � � � � � � � � � � � � �� � �� 2 2 2 0 4 0 01 1 � � � � �� t x a t c d d � � �� � � � � � � �� � � � � � � � � � � � � � 2 21 2 1 �� � � �� �t ax c t axp � � � � � . (12) A one-dimensional model equation describing the temperature distribution is derived from the energy equation for an ideal gas, see [5], � � � � � � � � � � � c t p v x v x v x v x V i i i j j i ij d d � � � � � � � � � � �2 3 � � � � � � � � � � � � � � � � � � � � � � � � v x v x v x x x i j ij i j j j � � � , (13) where �ij is the Kronecker delta, indices i, j, � go from 1 to 3 and vi represents three components of the acoustic velo- city vector. A one-dimensional form of Eq. (13) is obtained using the relation for one-dimensional divergence in an axi- symetrically shaped waveguide, see [1], � � � � � � � � � c t p r x r v r x r v r V d d � � � � �� � � � � �� � �� � � 2 2 2 2 24 3 1 2 2 2 � � � � � � �x r x y � �� � � � � , (14) where y is the spatial coordinate normal to the resonator wall. The last term of Eq. (14) describes the heat flow through the boundary layer to the resonator wall. For the temperature in the resonator cavity, we can write � � � � � �x y t x y t x tB M, , , , ,� � , where �B is the temper- ature in the boundary layer and �M is the mainstream temperature. It is easy to show, see [3], that � �B Ek k k tk y e� � � � � � � � ��� � � exp Prj j�� �0 (15) and thus � � � � � �� � �B y Ek k t k y k e t� ��� � � � � �� 0 0 0 Pr Pr j d d j 1 2 1 2 . (16) Integrating Eq. (14) with respect to the resonator cross- -section with help of the divergence theorem yields � � � � � � � � � c t p r x r v r x r vV d d � � � �� � � � � �� � �� � � � � � 2 2 2 2 24 3 1 � � � � � � � � � �� � � � � � � � � � �r x r x y2 2 � �d d= � , (17) where � is the resonator cross-section and � is its boundary. After substitution of relation (16) into Eq. (17) and calcul- ation of the integrals (acoustic quantities are assumed to be constant with respect to the integration domain), we obtain the model equation for the temperature distribution © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 9 Acta Polytechnica Vol. 42 No. 4/2002 � � � � � � � � � � � � � � �� � � � � � � � �� � � � 0 0 2 21 4 3 1 c t p p r x r x V d d � r x r x r x r x r 2 2 2 2 2 2 � � �� � � � � � � � � � � �� � �� � � � � � � � � Pr � � �0 1 2 1 2 � t . (18) Eq. (10) is solved numerically in the frequency domain, and driving acceleration a is assumed to be periodic. Owing to the numerical instability of Eq. (18) solved in the frequency domain, the mean temperature change is estimated from the thermodynamic state equation (it is assumed that all the heat generated in the resonator cavity due to viscous losses is conducted out through the resonator cavity walls) as � �� 1 0 0 � � � �� � � � � � p p � � . (19) 3 Numerical results and conclusions Fig. 1 shows the spatial distribution of the mean tempera- ture in the cylindrical, conical and bulb resonator due to the medium thermodynamic state change in an intensive sound field. The temperature was estimated using Eq. (19). The resonator driving acceleration a 3000 m�s�2, the driv- ing frequency agrees with the 1st natural frequency of the resonant cavities, the length of the resonators � 0.17 m, the radiuses of the resonators are: � cylindrical resonator – � r x � 0 01. m, 10 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 42 No. 4/2002 Fig. 1: Mean temperature distribution in a cylindrical, conical and bulb resonator � conical resonator – � r x x� �0 0056 0 268. . m, � bulb resonator – � � � r x x x� �0 003 3 0 005. sin exp .� � � m. It can be seen that the mean temperature is found near the pressure antinodes. Fig. 2 compares the acoustic velocity spectrum distribut- ion in a conical resonator where the influence of the temper- ature distribution is taken into account (dashed line) and where it is not taken into account (solid line). Fig. 3 compares the frequency characteristics for the same resonator. The numerical results show a slight resonant frequency shift and a waveform changeover if these mean temperature changes in the medium are taken into account. Fig. 4 shows an example of a thermoacoustic engine. The upper figure shows the mean temperature distribution, the bottom figure compares the acoustic velocity spectrum if the mean temperature distribution is taken into account (dashed © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 11 Acta Polytechnica Vol. 42 No. 4/2002 Fig. 2: Distribution of the acoustic velocity spectrum in conical resonator. Dashed line: temperature distribution is taken into account, solid line: temperature distribution is not taken into account. Fig. 3: Frequency characteristics of a conical resonator. Dashed line: temperature distribution is taken into account, solid line: temper- ature distribution is not taken into account. line) and if the mean temperature distribution is not taken into account (solid line). References [1] Ilinskii, Y. A., Lipkens, B., Lucas, T. S., Van Doren, T. W., Zabolotskaya, E. A.: Nonlinear standing waves in an acoustical resonator. J. Acoust. Soc. Am. Vol. 104, 1998, p. 2664–2674. [2] Bednařík, M., Červenka, M.: Nonlinear Waves in Resona- tors. Proc. of the 15th ISNA, Goettingen (Germany), 1999. [3] Chester, W.: Resonant oscillations in closed tubes. J. Fluid Mech., Vol. 18, 1964, p. 44–64. [4] Makarov, S., Ochmann, M.: Nonlinear and Thermovisc- ous Phenomena in Acoustics. Part I, ACUSTICA – Acta Acustica, Vol. 82, 1996, p. 579–605. [5] Blackstock, D. T.: Fundamentals of physical acoustics. New York: John Wiley & Sons, Inc., 2000, p. 77–84. Ing. Milan Červenka phone: +420 224 353 975 e-mail: cervenm3@feld.cvut.cz Dr. Ing. Michal Bednařík phone: +420 224 352 308 e-mail: bednarik@feld.cvut.cz Dr. Mgr. Petr Koníček phone: +420 224 352 329 e-mail: konicek@feld.cvut.cz Department of Physics Czech Technical University in Prague Faculty of Electrical Engineering Technická 2 166 27 Prague, Czech Republic 12 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 42 No. 4/2002 Fig. 4: Mean temperature distribution in cylindrical resonator due to external heating and cooling (upper figure), comparison of acous- tic velocity spectra in the resonator if the mean temperature distribution is taken into account (dashed line) and if the mean tem- perature distribution is not taken into account (solid line).