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Attenuation of sound waves in lattice gases
Physics Letters A 163 (1992) 392—396 North-Holland
PHYSICS LETTERS A
Attenuation of sound waves in lattice gases Paul Lavallée Physics Department, Université du Québec a Montréal, CP 8888, Montreal, Canada H3C 3P8 Received 29 July 1991; accepted for publication 28 January 1992 Communicated by D.D. Hoim
The theoretical expression for the attenuation of plane waves for a two-dimensional FHP lattice gas with zero average velocity is derived and compared with simulation results.
1. Introduction
2. Compressible gases. Wave equation in a medium of infinite extent
Lattice gases have been used to model incompressible flows obeying the Navier—Stokes equations in several fields related to hydrodynamics [1,2]. It was also shown early in their short history that lattice gases can be used to represent compressible waves [3]. As reported in ref. [41,lattice gases with FHP collision rules will show unphysical effects if the mean velocity of the gas is not zero. Simulation of a circular sound wave originating from a square “speaker” was reported by Margolus, Toffoli and Vichniac in 1986 [5]. More recently, a different type of model, capable of simulating wave propagation without attenuation, was proposed by Chen, Chen and Doolen [6]. The ability oflattice gases to deal easily with cornplex boundaries (lattice gases are used to simulate efficiently the flow of fluid through such complex boundaries as porous media [7]) could make them equally attractive to simulate progression of pressure waves through complex geometries. The simulation of sound waves in lattice gases has not been really explored until now. The present work is a first step in this direction. In section 2 we introduce the theoretical context in which sound waves can be simulated in lattice gases. Expressions for the attenuation of waves are derived. In section 3 the lattice gas experiment is explained. In section 4 simulation results are given and compared with theory.
392
The macrodynamical equations for mass and momentum for the evolution of a lattice gas (velocity U, density p) obeying the FHP collision rules are [3]
a1p+v~(pU)=o, ô1(pU) + V~P=V•S+higher order terms,
(1)
which are the continuity and the momentum equations. P and S are the momentum flux and the viscous stress tensors, respectively. Let us consider a gas with zero average particle yelocity and let u now represent the velocity fluctuations; the density is written as the sum of a constant part Po and a fluctuating part p. It was shown in ref. [31 that if eq. (1) is 1U, scaled according to r= er1, a>O, one obtains t=e’11, p=ë’p1, u=e’ ôp/8t+p 2Vp = 0, (2) 0 V~U = 0, Po ôu/t91 + c where c = ~ is the velocity of sound (for a twodimensional FHP gas with one rest particle). ii can be eliminated from these equations to give the wave equation 8
—
C2 V2p=
0.
(3)
On a longer time scale, if second order terms are considered with t=e2t~ in addition to the above scaling (with a> 1), the equations of motion now become
0375-9601 /92/s 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
Volume 163, number 5,6
PHYSICS LETTERS A
30 March 1992
2l~x]
ôp/ôt+p0V~u=0, p 2Vp 0ôu/81+c =pov{V2u+ [(D_2)/D]V(V.u)}+p
p~Aexp[i(w/c) (x—ct)] exp[
0CV(V.u),
— ~
(w/c)
rriA exp[i(w/c)(x—ct)] exp(—ax), (4)
where v is the dynamic strain viscosity and ~ is the (~is different from zero for adynamic gas withbulk restviscosity particles). In two dimensions D= 2 and (4) becomes
(9a) (9b)
where 21~ (10) a = ~(wIc) is the damping coefficient. The damped wave equation (9b) will be corn-
ôp/ôt+p 0 V~u = 0, 2Vp=p 2U+p Po 8u/0t+c 0vV 0~V(Vu). (5) (In three dimensions the term PoCV (V.u) is replaced by the more familar expression p0(~+4v)V(V.u).) Since the gradient of a scalar function f is entirely longitudinal, VxVf=O, foranyf the equation ofmotion can be split into two separate equations, one relating p to the longitudinal part U5 of u (for which V x u5= 0), the other giving the evolution of the transverse part u~of u (for which V~u~= 0). The transverse part of the solution, u~,is not related to pressure waves; it is important only near boundaries. Making use of the vector relation 2u = V(Vu) VXV X U, V eq. (5) can be decomposed into a longitudinal part and a fast decaying transverse part [8] 2Vp=p 2u Po 8u5/ôt+c 0(v+~)V 5, 8u1/3t=
—
VVXV
xu~.
(6a)
(6b)
We eliminate 115 from eq. (6a) by differentiating (6a) with respect to the space coordinates, and by substituting the appropriate terms from the timedifferentiated continuity equation. The resulting equation is 2V2p= ci,.. ôV2p/8t, (7) c where —
1~,= (~+v ) Ic.
(8)
Eq. (7) reduces to the harmonic wave equation (3) if the r.h.s. term is zero (i.e. zero viscosity). The r.h.s. term is a damping of the wave motion. If this term is small (ci,., 0.1 for a FHP III gas at a density d= 0.5), the solution to eq. (7) can be considered as a decaying sinusoidal wave [8],
pared with the results of simulations.
3. The lattice gas experiment The experiment simulates a pressure transducer vibrating in a rectangular cavity (Xm~= 320, Ymax = 256 lattice units). Particles colliding with the walls are reflected specularly. The pressure transducer injects particles with pressure ranging sinusoidally from P+ d.P to P— dP with a selected period T (injection ofpulses was also tried). The shape of the pressure “transducer” is designed as to generate either or plane in the so first case, particles arecircular injected insidewaves: a circular region of radius R, centered at position X 0, Y0 within the cavity (injection inside a square region was also tried, without any significant difference). Plane waves are excited by injecting particles in a “column”, two particles wide, located at y=0 for a wave propagating in the x direction, or at x=0 for a wave in the y direction. Sound propagation in lattice gases is inherently noisy and we had to use a method to reduce the noise; for a plane wave travelling in the x direction, for exvalue ofthe ample, x was average computed,but number ofa region particles near forthe a given walls was excluded from this average so as to eliminate the slowing effect of the boundaries on the wave. The sequence of values of the density was then plotted as a function of x. In a FHP lattice gas, the pressure level is proportional to the particle density and is identified in the present context with the particle density.
393
Volume 163, number 5,6
PHYSICS LETTERSA
4. Simulation results. Damping coefficient of a plane wave
30 March 1992
______________________________________ 5.6
Expression (10) forthe attenuation of plane waves, ~ (co/c) 2/,, together with (8) predicts that attenuation is proportional to the sum of shear and bulk viscosities, and that it increases with the square of the stimulation frequency. The expression for 1., is minimum at a density d= 0.5 and all our simulations were run at this density except for a few where the
‘
a=
value of!,, was tested. The results appear Fig. 1 shows a concentration map for a wave with a period T= 40. A density larger than the average at a given point is indicated by a white dot. Propagation is from left to right. Attenuation is visible as the wave progresses to the right of the picture. In order to quantify the attenuation, the particle density on a horizontal line is traced in fig. 2 as a function of distance. The attenuation of this density profile is now clearly visible and measurable. Figs. 3 and 4 ~
~ ~ ~ ~ 2.8 ~
‘~
•
r~Iw __ . • ~j
:~r1~w ~ ~
~17~VWr~ . L~ ‘:.——~~~ ~‘. ~
~ ~ ~~t~.U~~.gj
~
~
-
• .~.
Fig. I. Propagation of a plane wave in a cavity (the cavity prevents the wave from going in the right to left direction). The penod ofthe wave is 40. The density varies sinusoidally from d=0.3 to d=0.7 in figs. 1—5.
394
,
400
I
Distance (lattice units) Fig. 2. Density (pressure) profile of a wave for a period T= 40, density d=0.5. The spikes near x=0 and X=XmaX are artefacts dueto the averaging technique.
5.6
______________________________________ ~ ~ ~ 4_~4
I
o
t2.8 0 1.4 0
~
I
I 200
400
Fig. 3. Density (pressure) Distance profile (lattice of a wave units) for with a period T= 20, density show the d=0.5. density profiles for waves periods
T= 20 and T= 80 respectively. Fig. 5 shows the attenuation coefficients a for different frequencies and densities. The results of these simulations are summarized in table 1.
Volume 163, number 5,6
PHYSICS LETTERS A
30 March 1992
Table 1 Comparison between theoretical values of the attenuation coefficient and the values obtained by simulation (d is the particle density, a,,, is the theoretical value of the attenuation coefficient and a,im is the value of the attenuation coefficient obtained from the simulation).
d
T=20
0.25 0.50
T=40
a,,,
a,im
—
—
0.0213
0.0222
a,,,
a,jm
a,
0.0075 0.0053
0.0082 0.0057
0.00133
1,
a,jm —
0.00168
The results of the simulation agree quite well with the theoretical values, taking into account the noisy nature of the simulations. The large attenuation at high frequencies is clearly put in evidence and the dependence on density is also established. The attenuation factor between waves separated by one octave should be 4.0. It is 3.9 and 3.4 respectively for the waves tested. In order to obtain more accurate results one would have to use a perfectly absorbing wall (reflecting particles in random directions) and average over several periods of the wave.
8)~
j3.. a,
~ 2a,
z =
—
T=80
1—
5. Conclusion
-
0
202
400
Distance (lattice units)
Fig. 4. Density (pressure) profile of a wave for a period T= 80, density d=0.5.
__________________________________ 0•
0100167
0 —
0 00570
T=40. d=O.5
—
4. 0
T20 • d~05
a ~ 0.02247 T—40. d—0.25
0 0081.9
~
•
100 200 300 distance (lattice units)
400
Fig. 5. Attenuation coefficients a for a number ofwaves of different periods and particle densities.
Pressure waves in a lattice gas at rest are shown theoretically to attenuate like those of a damped harmonic oscillator if the pressure variation is not too large. Consequently, the attenuation coefficient of these waves is proportional to the product of the viscosity and w~.The results of the simulations show that attenuation of pressure waves decreases with the square of the period in accordance with theoretical results. The value of the measured attenuation coefficients is also shown to agree reasonably well with the calculated theoretical values.
References [1] G.D. Doolen,
ed., Proc. Workshop on Large nonlinear
systems. Lattice gas methods for partial differential equations (Addison-Wesley, Reading, MA, 1990). [2] GD. Doolen, ed., Proc. NATO Advanced Research Workshop on Lattice gas methods for PDE’s, Physica D 47 (1991). [3] U. Frisch, D. d’Humières, B. Hasslacher, P. Lallemand, Y. Pomeau and J.P. Rivet, Complex Syst. 1 (1987) 648.
395
Volume 163, number 5,6
PHYSICS LETTERSA
[4] H. Chen, S. Chen and GD. Doolen, Phys. Lett. A 140 (1989) 161. [51N. Margolus, T. Toffoli and G. Vichniac, Phys. Rev. Lett. 56 (1986) 1691. [6] H. Chen, S. Chen and G.D. Doolen, Complex Syst. 2 (1988) 259.
396
30 March 1992
[7] D. Rothman, Geophysics 53 (1988) 509. [8] P. Morse and U. Ingard, Theoretical acoustics (McGraw-Hill, New York, 1968).
PHYSICS LETTERS A
Attenuation of sound waves in lattice gases Paul Lavallée Physics Department, Université du Québec a Montréal, CP 8888, Montreal, Canada H3C 3P8 Received 29 July 1991; accepted for publication 28 January 1992 Communicated by D.D. Hoim
The theoretical expression for the attenuation of plane waves for a two-dimensional FHP lattice gas with zero average velocity is derived and compared with simulation results.
1. Introduction
2. Compressible gases. Wave equation in a medium of infinite extent
Lattice gases have been used to model incompressible flows obeying the Navier—Stokes equations in several fields related to hydrodynamics [1,2]. It was also shown early in their short history that lattice gases can be used to represent compressible waves [3]. As reported in ref. [41,lattice gases with FHP collision rules will show unphysical effects if the mean velocity of the gas is not zero. Simulation of a circular sound wave originating from a square “speaker” was reported by Margolus, Toffoli and Vichniac in 1986 [5]. More recently, a different type of model, capable of simulating wave propagation without attenuation, was proposed by Chen, Chen and Doolen [6]. The ability oflattice gases to deal easily with cornplex boundaries (lattice gases are used to simulate efficiently the flow of fluid through such complex boundaries as porous media [7]) could make them equally attractive to simulate progression of pressure waves through complex geometries. The simulation of sound waves in lattice gases has not been really explored until now. The present work is a first step in this direction. In section 2 we introduce the theoretical context in which sound waves can be simulated in lattice gases. Expressions for the attenuation of waves are derived. In section 3 the lattice gas experiment is explained. In section 4 simulation results are given and compared with theory.
392
The macrodynamical equations for mass and momentum for the evolution of a lattice gas (velocity U, density p) obeying the FHP collision rules are [3]
a1p+v~(pU)=o, ô1(pU) + V~P=V•S+higher order terms,
(1)
which are the continuity and the momentum equations. P and S are the momentum flux and the viscous stress tensors, respectively. Let us consider a gas with zero average particle yelocity and let u now represent the velocity fluctuations; the density is written as the sum of a constant part Po and a fluctuating part p. It was shown in ref. [31 that if eq. (1) is 1U, scaled according to r= er1, a>O, one obtains t=e’11, p=ë’p1, u=e’ ôp/8t+p 2Vp = 0, (2) 0 V~U = 0, Po ôu/t91 + c where c = ~ is the velocity of sound (for a twodimensional FHP gas with one rest particle). ii can be eliminated from these equations to give the wave equation 8
—
C2 V2p=
0.
(3)
On a longer time scale, if second order terms are considered with t=e2t~ in addition to the above scaling (with a> 1), the equations of motion now become
0375-9601 /92/s 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
Volume 163, number 5,6
PHYSICS LETTERS A
30 March 1992
2l~x]
ôp/ôt+p0V~u=0, p 2Vp 0ôu/81+c =pov{V2u+ [(D_2)/D]V(V.u)}+p
p~Aexp[i(w/c) (x—ct)] exp[
0CV(V.u),
— ~
(w/c)
rriA exp[i(w/c)(x—ct)] exp(—ax), (4)
where v is the dynamic strain viscosity and ~ is the (~is different from zero for adynamic gas withbulk restviscosity particles). In two dimensions D= 2 and (4) becomes
(9a) (9b)
where 21~ (10) a = ~(wIc) is the damping coefficient. The damped wave equation (9b) will be corn-
ôp/ôt+p 0 V~u = 0, 2Vp=p 2U+p Po 8u/0t+c 0vV 0~V(Vu). (5) (In three dimensions the term PoCV (V.u) is replaced by the more familar expression p0(~+4v)V(V.u).) Since the gradient of a scalar function f is entirely longitudinal, VxVf=O, foranyf the equation ofmotion can be split into two separate equations, one relating p to the longitudinal part U5 of u (for which V x u5= 0), the other giving the evolution of the transverse part u~of u (for which V~u~= 0). The transverse part of the solution, u~,is not related to pressure waves; it is important only near boundaries. Making use of the vector relation 2u = V(Vu) VXV X U, V eq. (5) can be decomposed into a longitudinal part and a fast decaying transverse part [8] 2Vp=p 2u Po 8u5/ôt+c 0(v+~)V 5, 8u1/3t=
—
VVXV
xu~.
(6a)
(6b)
We eliminate 115 from eq. (6a) by differentiating (6a) with respect to the space coordinates, and by substituting the appropriate terms from the timedifferentiated continuity equation. The resulting equation is 2V2p= ci,.. ôV2p/8t, (7) c where —
1~,= (~+v ) Ic.
(8)
Eq. (7) reduces to the harmonic wave equation (3) if the r.h.s. term is zero (i.e. zero viscosity). The r.h.s. term is a damping of the wave motion. If this term is small (ci,., 0.1 for a FHP III gas at a density d= 0.5), the solution to eq. (7) can be considered as a decaying sinusoidal wave [8],
pared with the results of simulations.
3. The lattice gas experiment The experiment simulates a pressure transducer vibrating in a rectangular cavity (Xm~= 320, Ymax = 256 lattice units). Particles colliding with the walls are reflected specularly. The pressure transducer injects particles with pressure ranging sinusoidally from P+ d.P to P— dP with a selected period T (injection ofpulses was also tried). The shape of the pressure “transducer” is designed as to generate either or plane in the so first case, particles arecircular injected insidewaves: a circular region of radius R, centered at position X 0, Y0 within the cavity (injection inside a square region was also tried, without any significant difference). Plane waves are excited by injecting particles in a “column”, two particles wide, located at y=0 for a wave propagating in the x direction, or at x=0 for a wave in the y direction. Sound propagation in lattice gases is inherently noisy and we had to use a method to reduce the noise; for a plane wave travelling in the x direction, for exvalue ofthe ample, x was average computed,but number ofa region particles near forthe a given walls was excluded from this average so as to eliminate the slowing effect of the boundaries on the wave. The sequence of values of the density was then plotted as a function of x. In a FHP lattice gas, the pressure level is proportional to the particle density and is identified in the present context with the particle density.
393
Volume 163, number 5,6
PHYSICS LETTERSA
4. Simulation results. Damping coefficient of a plane wave
30 March 1992
______________________________________ 5.6
Expression (10) forthe attenuation of plane waves, ~ (co/c) 2/,, together with (8) predicts that attenuation is proportional to the sum of shear and bulk viscosities, and that it increases with the square of the stimulation frequency. The expression for 1., is minimum at a density d= 0.5 and all our simulations were run at this density except for a few where the
‘
a=
value of!,, was tested. The results appear Fig. 1 shows a concentration map for a wave with a period T= 40. A density larger than the average at a given point is indicated by a white dot. Propagation is from left to right. Attenuation is visible as the wave progresses to the right of the picture. In order to quantify the attenuation, the particle density on a horizontal line is traced in fig. 2 as a function of distance. The attenuation of this density profile is now clearly visible and measurable. Figs. 3 and 4 ~
~ ~ ~ ~ 2.8 ~
‘~
•
r~Iw __ . • ~j
:~r1~w ~ ~
~17~VWr~ . L~ ‘:.——~~~ ~‘. ~
~ ~ ~~t~.U~~.gj
~
~
-
• .~.
Fig. I. Propagation of a plane wave in a cavity (the cavity prevents the wave from going in the right to left direction). The penod ofthe wave is 40. The density varies sinusoidally from d=0.3 to d=0.7 in figs. 1—5.
394
,
400
I
Distance (lattice units) Fig. 2. Density (pressure) profile of a wave for a period T= 40, density d=0.5. The spikes near x=0 and X=XmaX are artefacts dueto the averaging technique.
5.6
______________________________________ ~ ~ ~ 4_~4
I
o
t2.8 0 1.4 0
~
I
I 200
400
Fig. 3. Density (pressure) Distance profile (lattice of a wave units) for with a period T= 20, density show the d=0.5. density profiles for waves periods
T= 20 and T= 80 respectively. Fig. 5 shows the attenuation coefficients a for different frequencies and densities. The results of these simulations are summarized in table 1.
Volume 163, number 5,6
PHYSICS LETTERS A
30 March 1992
Table 1 Comparison between theoretical values of the attenuation coefficient and the values obtained by simulation (d is the particle density, a,,, is the theoretical value of the attenuation coefficient and a,im is the value of the attenuation coefficient obtained from the simulation).
d
T=20
0.25 0.50
T=40
a,,,
a,im
—
—
0.0213
0.0222
a,,,
a,jm
a,
0.0075 0.0053
0.0082 0.0057
0.00133
1,
a,jm —
0.00168
The results of the simulation agree quite well with the theoretical values, taking into account the noisy nature of the simulations. The large attenuation at high frequencies is clearly put in evidence and the dependence on density is also established. The attenuation factor between waves separated by one octave should be 4.0. It is 3.9 and 3.4 respectively for the waves tested. In order to obtain more accurate results one would have to use a perfectly absorbing wall (reflecting particles in random directions) and average over several periods of the wave.
8)~
j3.. a,
~ 2a,
z =
—
T=80
1—
5. Conclusion
-
0
202
400
Distance (lattice units)
Fig. 4. Density (pressure) profile of a wave for a period T= 80, density d=0.5.
__________________________________ 0•
0100167
0 —
0 00570
T=40. d=O.5
—
4. 0
T20 • d~05
a ~ 0.02247 T—40. d—0.25
0 0081.9
~
•
100 200 300 distance (lattice units)
400
Fig. 5. Attenuation coefficients a for a number ofwaves of different periods and particle densities.
Pressure waves in a lattice gas at rest are shown theoretically to attenuate like those of a damped harmonic oscillator if the pressure variation is not too large. Consequently, the attenuation coefficient of these waves is proportional to the product of the viscosity and w~.The results of the simulations show that attenuation of pressure waves decreases with the square of the period in accordance with theoretical results. The value of the measured attenuation coefficients is also shown to agree reasonably well with the calculated theoretical values.
References [1] G.D. Doolen,
ed., Proc. Workshop on Large nonlinear
systems. Lattice gas methods for partial differential equations (Addison-Wesley, Reading, MA, 1990). [2] GD. Doolen, ed., Proc. NATO Advanced Research Workshop on Lattice gas methods for PDE’s, Physica D 47 (1991). [3] U. Frisch, D. d’Humières, B. Hasslacher, P. Lallemand, Y. Pomeau and J.P. Rivet, Complex Syst. 1 (1987) 648.
395
Volume 163, number 5,6
PHYSICS LETTERSA
[4] H. Chen, S. Chen and GD. Doolen, Phys. Lett. A 140 (1989) 161. [51N. Margolus, T. Toffoli and G. Vichniac, Phys. Rev. Lett. 56 (1986) 1691. [6] H. Chen, S. Chen and G.D. Doolen, Complex Syst. 2 (1988) 259.
396
30 March 1992
[7] D. Rothman, Geophysics 53 (1988) 509. [8] P. Morse and U. Ingard, Theoretical acoustics (McGraw-Hill, New York, 1968).