^{1}, Kyungmin Baik

^{1}and Timothy G. Leighton

^{1,a)}

### Abstract

This paper uses a Finite Element Method(FEM) to compare predictions of the attenuation and sound speeds of acoustic modes in a fluid-filled pipe with those of the analytical model presented in the first paper in this series. It explains why, when the predictions of the earlier paper were compared with experimental data from a water-filled PMMA pipe, the uncertainties and agreement for attenuation data were worse than those for sound speed data. Having validated the FEM approach in this way, the versatility of FEM is thereafter demonstrated by modeling two practical applications which are beyond the analysis of the earlier paper. These applications model propagation in the mercury-filled steel pipework of the Spallation Neutron Source at the Oak Ridge National Laboratory (Tennessee), and in a long-standing design for acoustic sensors for use on planetary probes. The results show that strong coupling between the fluid and the solid walls means that erroneous interpretations are made of the data if they assume that the sound speed and attenuation in the fluid in the pipe are the same as those that would be measured in an infinite volume of identical fluid, assumptions which are common when such data have previously been interpreted.

This work is supported by the Oak Ridge National Laboratory (ORNL), TN (ORNL is managed by UT-Battelle, LLC, under contract DE-AC05-00OR22725 for the U.S. Department of Energy) and by the UK Science and Technology Research Council Rutherford Appleton Laboratory (Principle Investigator: T.G.L.). The authors are very grateful to Bernie Riemer and Mark Wendel of ORNL, and Chris Densham, Ottone Caretta, Tristan Davenne, and Tim Broome of RAL, for advice and discussions.

I. INTRODUCTION

II. FORMALISM

III. SPALLATION NEUTRON SOURCE

IV. ACOUSTIC PLANETARY EXPLORATION SENSOR

V. CONCLUSION

### Key Topics

- Wave attenuation
- 37.0
- Acoustic sensing
- 21.0
- Speed of sound
- 21.0
- Venus
- 18.0
- Acoustical measurements
- 16.0

## Figures

Geometrical dimensions of the water-filled PMMA pipe (not drawn in scale).

Geometrical dimensions of the water-filled PMMA pipe (not drawn in scale).

(Color online) Waveform and spectrum of a typical impulse signal. Throughout this whole paper the constant for normalization relating to the “normalized acoustic pressure” is unchanged.

(Color online) Waveform and spectrum of a typical impulse signal. Throughout this whole paper the constant for normalization relating to the “normalized acoustic pressure” is unchanged.

(Color online) Time history map of for the signal of Fig. 2 (the study was carried out in a water-filled PMMA pipe whose geometrical dimensions are shown in Fig. 1). Color scale shows the normalized acoustic pressure [as in Fig. 2(a)] in linear representation. Signals are calculated for the center of the pipe cross section, using the frequency domain method. See supplementary material in Ref. 55 for a movie of a pulse signal propagating though water filled PMMA pipe, calculated using the alterative time-transient method. It can also be found at the web page http://www.isvr.soton.ac.uk/fdag/PIPE_DEMO/index.htm (last viewed 23 March 2011).

(Color online) Time history map of for the signal of Fig. 2 (the study was carried out in a water-filled PMMA pipe whose geometrical dimensions are shown in Fig. 1). Color scale shows the normalized acoustic pressure [as in Fig. 2(a)] in linear representation. Signals are calculated for the center of the pipe cross section, using the frequency domain method. See supplementary material in Ref. 55 for a movie of a pulse signal propagating though water filled PMMA pipe, calculated using the alterative time-transient method. It can also be found at the web page http://www.isvr.soton.ac.uk/fdag/PIPE_DEMO/index.htm (last viewed 23 March 2011).

(Color online) The time domain waveform at different positions for the input shown in Fig. 2(a) (the study was carried out in a water-filled PMMA pipe whose geometrical dimensions are shown in Fig. 1).

(Color online) The time domain waveform at different positions for the input shown in Fig. 2(a) (the study was carried out in a water-filled PMMA pipe whose geometrical dimensions are shown in Fig. 1).

(Color online) 2D spectrum with axis of normalized phase velocity against normalized wave number (the study was carried out in a water-filled PMMA pipe whose geometrical dimensions are shown in Fig. 1). Since the color scale value is the output of a 2D FFT of data such as that found in Fig. 2, it shows (in dB) the “normalized acoustic pressure per Hz per meter,” where 0 dB corresponds to the situation where the original output of 2D FFT equals 1 Hz^{−1} m^{−1}. Circles, triangles, squares, and crosses represent ET1, ET2, ET3, and ET4 modes, respectively. White symbols correspond to theoretical results and black symbols correspond to experimental results.

(Color online) 2D spectrum with axis of normalized phase velocity against normalized wave number (the study was carried out in a water-filled PMMA pipe whose geometrical dimensions are shown in Fig. 1). Since the color scale value is the output of a 2D FFT of data such as that found in Fig. 2, it shows (in dB) the “normalized acoustic pressure per Hz per meter,” where 0 dB corresponds to the situation where the original output of 2D FFT equals 1 Hz^{−1} m^{−1}. Circles, triangles, squares, and crosses represent ET1, ET2, ET3, and ET4 modes, respectively. White symbols correspond to theoretical results and black symbols correspond to experimental results.

(Color online) Normalized attenuation plotted against *k* _{1} *b* (the study was carried out in a water-filled PMMA pipe whose geometrical dimensions are shown in Fig. 1). Here Im[*q* _{0m}] is the imaginary part of wave number which is used in Ref. 7. The solid and dashed lines represent ET1 and ET2 modes, respectively, from the analytical model. The triangles and circles represent ET1 and ET2 modes from the simulation.

(Color online) Normalized attenuation plotted against *k* _{1} *b* (the study was carried out in a water-filled PMMA pipe whose geometrical dimensions are shown in Fig. 1). Here Im[*q* _{0m}] is the imaginary part of wave number which is used in Ref. 7. The solid and dashed lines represent ET1 and ET2 modes, respectively, from the analytical model. The triangles and circles represent ET1 and ET2 modes from the simulation.

(Color online) A 2D spectrum with axis of normalized phase velocity against normalized wave number for the steel pipe (wall thickness 0.655 cm) containing mercury to an inner diameter of 12.82 cm. The color scale value (see Fig. 5) is the output of a 2D FFT and shows (in dB) the “normalized acoustic pressure per Hz per meter,” where 0 dB corresponds to the situation where the original output of 2D FFT equals 1 Hz^{−1} m^{−1}. Circles, triangles, squares, and crosses represent ET1, ET2, ET3, and ET4 modes, respectively.

(Color online) A 2D spectrum with axis of normalized phase velocity against normalized wave number for the steel pipe (wall thickness 0.655 cm) containing mercury to an inner diameter of 12.82 cm. The color scale value (see Fig. 5) is the output of a 2D FFT and shows (in dB) the “normalized acoustic pressure per Hz per meter,” where 0 dB corresponds to the situation where the original output of 2D FFT equals 1 Hz^{−1} m^{−1}. Circles, triangles, squares, and crosses represent ET1, ET2, ET3, and ET4 modes, respectively.

(Color online) Still (at *t* = 2.5 ms) taken from a movie simulating the propagation of an acoustic pulse in ORNL’s SNS Target Test Facility (TTF). As in the simulation, the SNS TTF pipeline consists of a target section (the chamber at the left of the simulation, where the proton beam impacts the mercury). Mercury is pumped into the target section through the two outer pipelines (each of which contains an offset bend to raise the level of the mercury by 23 inches to the height of the target section). The return flow is via the central pipeline. In the existing test loop, the return pipe also contains an offset bend matching those on the inflow pipes. This particular simulation was undertaken to test the possibility of keeping the return flow at the height of the target section in order to facilitate propagation of acoustic pulses along the return pipe toward the target section, in order to measure the bubble population there (the topic of a future paper). See supplementary material in Ref. 55 for a movie of this simulation. It can also be found at the web page http://www.isvr.soton.ac.uk/fdag/PIPE_DEMO/index.htm (last viewed 23 March 2011).

(Color online) Still (at *t* = 2.5 ms) taken from a movie simulating the propagation of an acoustic pulse in ORNL’s SNS Target Test Facility (TTF). As in the simulation, the SNS TTF pipeline consists of a target section (the chamber at the left of the simulation, where the proton beam impacts the mercury). Mercury is pumped into the target section through the two outer pipelines (each of which contains an offset bend to raise the level of the mercury by 23 inches to the height of the target section). The return flow is via the central pipeline. In the existing test loop, the return pipe also contains an offset bend matching those on the inflow pipes. This particular simulation was undertaken to test the possibility of keeping the return flow at the height of the target section in order to facilitate propagation of acoustic pulses along the return pipe toward the target section, in order to measure the bubble population there (the topic of a future paper). See supplementary material in Ref. 55 for a movie of this simulation. It can also be found at the web page http://www.isvr.soton.ac.uk/fdag/PIPE_DEMO/index.htm (last viewed 23 March 2011).

(Color online) A frame (at t = 0.2 ms after the emission of the pulse by the source) taken from a movie showing an impulse signal generated at the source end of the spiral shape planet exploration sensor and propagating along it. The hatched arm of the spiral is solid, the unhatched arm is gaseous. The inset shows the geometrical dimensions of the channel as relate to this 2D simulation (such that the height of the channel, shown as a dashed line) is not included in the simulation. See supplementary material in Ref. 55 for a movie of this simulation. It can also be found at the web page http://www.isvr.soton.ac.uk/fdag/PIPE_DEMO/index.htm (last viewed 23 March 2011).

(Color online) A frame (at t = 0.2 ms after the emission of the pulse by the source) taken from a movie showing an impulse signal generated at the source end of the spiral shape planet exploration sensor and propagating along it. The hatched arm of the spiral is solid, the unhatched arm is gaseous. The inset shows the geometrical dimensions of the channel as relate to this 2D simulation (such that the height of the channel, shown as a dashed line) is not included in the simulation. See supplementary material in Ref. 55 for a movie of this simulation. It can also be found at the web page http://www.isvr.soton.ac.uk/fdag/PIPE_DEMO/index.htm (last viewed 23 March 2011).

(Color online) Phase velocity measured by planet sensor (straight pipe). Circles, squares, triangles, and diamonds represent idealized rigid boundary, steel, aluminum, and carbon fiber, respectively. The open symbols correspond to air on the Earth and filled symbols correspond to gas on Venus.

(Color online) Phase velocity measured by planet sensor (straight pipe). Circles, squares, triangles, and diamonds represent idealized rigid boundary, steel, aluminum, and carbon fiber, respectively. The open symbols correspond to air on the Earth and filled symbols correspond to gas on Venus.

(Color online) Plots of the 2D spectrum with axis of phase velocity against frequency for straight pipes with 1 cm thick walls of (a) aluminum and (b) carbon fiber containing gas on Venus to an inner diameter of 1 cm. The color scale value which is the output of a 2D FFT shows (in dB) the “normalized acoustic pressure per Hz per meter,” where 0 dB corresponds to the situation where the original output of 2D FFT equals 1 Hz^{−1} m^{−1}.

(Color online) Plots of the 2D spectrum with axis of phase velocity against frequency for straight pipes with 1 cm thick walls of (a) aluminum and (b) carbon fiber containing gas on Venus to an inner diameter of 1 cm. The color scale value which is the output of a 2D FFT shows (in dB) the “normalized acoustic pressure per Hz per meter,” where 0 dB corresponds to the situation where the original output of 2D FFT equals 1 Hz^{−1} m^{−1}.

(Color online) Simulated resulted for measurements of ground-level atmospheres by straight and spiral pipe sensors where the solid walls are made of aluminum. (a) Phase velocity as a function of frequency, as measured by a spiral sensor having the dimensions of Fig. 9. Circles, squares, and triangles represent idealized rigid boundary, steel and aluminum, respectively. The open symbols correspond to the ground-level atmosphere on Earth and filled symbols correspond to that on Venus. (b) The attenuation per unit length, plotted against frequency, as if measured and averaged from 500 microphones placed equidistantly along the axis of the gas column. The solid line shows the input attenuation coefficient of gas on Venus. Circles and stars represent the results measured by using straight and spiral shape sensors, respectively.

(Color online) Simulated resulted for measurements of ground-level atmospheres by straight and spiral pipe sensors where the solid walls are made of aluminum. (a) Phase velocity as a function of frequency, as measured by a spiral sensor having the dimensions of Fig. 9. Circles, squares, and triangles represent idealized rigid boundary, steel and aluminum, respectively. The open symbols correspond to the ground-level atmosphere on Earth and filled symbols correspond to that on Venus. (b) The attenuation per unit length, plotted against frequency, as if measured and averaged from 500 microphones placed equidistantly along the axis of the gas column. The solid line shows the input attenuation coefficient of gas on Venus. Circles and stars represent the results measured by using straight and spiral shape sensors, respectively.

## Tables

The densities and wave speeds that are used as input parameters for the simulations of Spallation Neutron Source pipelines (specific acoustic impedances for compressional waves, (and, for solids, the equivalent for shear waves, ) are also shown, though these are not input parameters).

The densities and wave speeds that are used as input parameters for the simulations of Spallation Neutron Source pipelines (specific acoustic impedances for compressional waves, (and, for solids, the equivalent for shear waves, ) are also shown, though these are not input parameters).

The densities and wave speeds that are used as input parameters for the simulations of an acoustic planet sensor (specific acoustic impedances for compressional waves (and, for solids, the equivalent for shear waves) are also shown for the isotropic materials, though these are not input parameters).

The densities and wave speeds that are used as input parameters for the simulations of an acoustic planet sensor (specific acoustic impedances for compressional waves (and, for solids, the equivalent for shear waves) are also shown for the isotropic materials, though these are not input parameters).

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