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In the previous investigations of the vibroacoustic characteristics of a submerged cylindrical shell in a flow field, the fluid viscosity was usually ignored. In this paper, the effect of fluid viscosity on the characteristics of vibration power flow in an infinite circular cylindrical shell immersed in a viscous acoustic medium is studied. Flügge’s thin shell theory for an isotropic, elastic, and thin cylindrical shell is employed to obtain the motion equations of the structure under circumferential-distributed line force. Together with the wave equations for the viscous flow field as well as continuity conditions at the interface, the vibroacoustic equation of motion in the coupled system is derived. Numerical analysis based on the additional-damping numerical integral method and ten-point Gaussian integral method is conducted to solve the vibroacoustic coupling equation with varying levels of viscosity. Then, the variation of the input power flow against the nondimensional axial wave number in the coupled system with different circumferential mode numbers is discussed in detail. It is found that the influence of fluid viscosity on the vibroacoustic coupled system is mainly concentrated in the low-frequency band, which is shown as the increase of the crest number and amplitude of the input power flow curves.

Much research has been conducted on the analysis of the vibroacoustic characteristics of an infinite elastic thin cylindrical shell submerged in an acoustic medium. One of the earliest was done by Junger, who found that the existence of entrained water could greatly reduce the natural frequency of the shell due to added-mass effects and showed that the dynamic behavior of the structure was mainly affected in the low-frequency band [

It should be noted that the surrounding fluid was generally considered to be inviscid in the previous research. However, fluid viscosity is inevitable in practice and the ignorance of its influence on the vibroacoustic behavior of the submerged cylindrical shells may result in large inaccuracy in certain circumstances.

Amabili and Garziera investigated the effect of steady viscous forces on vibrations of shells with nonuniform constraints, added masses, elastic bed, and prestress conveying or immersed in annular axial flow and found that the effect of viscous loads can be considered stabilizing if compared with results computed for shells conveying inviscid flow having the same outlet pressure. Besides, the contrary is obtained if a comparison is made with the results computed for shells having the same inlet pressure [

Therefore, considerations of the influence of fluid viscosity can better reflect reality and thus provide a more accurate theoretical model. There is also little work done regarding the vibroacoustic characteristics of an infinite circular cylindrical shell submerged in viscous fluids.

As an attempt to address these issues, this paper investigates the vibroacoustic characteristics of an isotropic, elastic, and thin cylindrical shell coupled with compressible fluids taking into account the viscosity. The acoustic wave equations of the coupled system considering hydrostatic pressure are derived based on Flügge’s thin shell theory and the linearized equations of continuity as well as the Navier–Stokes equations. Then, the pertinent boundary conditions are employed to obtain the vibroacoustic coupled equation of motion of the coupled system under circumferential-distributed line force. After that, the numerical algorithms are developed to solve the equations with consideration of varying fluid viscosity to obtain the curves of the input power flow versus circular frequency with different circumferential mode numbers. Based on the results, the influence of fluid viscosity on the vibroacoustic characteristics of the coupled system is analyzed.

The geometry and the coordinate system are shown in Figure

Coordinate system and circumferential mode shapes.

Assume that a circular cosine-distributed harmonic force is applied at the section

It is assumed that the material of the thin cylindrical shell is isotropic and elastic, and the thickness of the shell is much smaller compared with its mean radius. The governing equations of motion are obtained based on Flügge’s thin shell theory, which can be written as [

Using a wave propagation approach, the solution of equation (

Substituting equation (

The fluid is assumed to be viscous, nonheat-conducting, compressible, and isotropic and, thus, the acoustic wave equation is valid. The linearized equation of continuity is

And the Navier–Stokes equation is

For a barotropic fluid, the linear equation of state is

Wave propagation and the consequent vibration characteristics are focused on in this paper, and the velocity vector _{f} at any point in the flow field is much smaller compared to the wave and sound velocity _{f}. Assuming the dimension of _{f} is ^{2}. Therefore, the product of any term involving fluid velocity components can be regarded as an infinitesimal of a higher order. With infinitesimals of higher order ignored, the total differential terms in the Navier–Stokes equation can be simplified, such as

Therefore, a partial derivative about

Then, equations (

Ignoring the higher-order terms in equation (_{f}, which is linearized [

According to the Helmholtz decomposition theorem, the velocity fields can be resolved as the superposition of longitudinal and transverse vector components:

Introducing the above decomposition into equation (

If the wave is assumed to be monochromatic, the solutions are expected to be of the following form [

Incorporate the assumptions above in equation (

In the cylindrical coordinate, the solutions can be expressed as follows [

Now, considering the basic field equations in cylindrical coordinates, the velocity components of the waves in

Similarly, the stress components in the fluid are [

With the harmonic terms ignored, a substitution of equation (

The stress components in the fluid derived using the wave propagation approach serve as the equivalent loads on the thin-walled cylindrical shell applied by the medium in the directions of _{x}, _{θ}, and _{r}, rather than the force dimension, is taken.

According to the motion coordination conditions of the outer surface of the shell, the velocities in the three directions must be continuous at the fluid-shell boundary, leading to the continuous boundary conditions:

A matrix equation can be obtained by substituting equation (

Substituting equation (

Then, the velocity continuous conditions lead to a matrix equation as follows:

After some manipulations, it follows that

The equation of motion of the coupled system can be obtained by combining equation (

In order to introduce the external excitation force into the motion equations, Fourier transform is needed for the force and shell displacement. The Fourier transform and that of differential form of variable function used in the derivation and calculation of the relevant formulas in this paper are defined, respectively, as follows:

Therefore, the external force and displacement components have the form of

Substituting equations (

Assuming that

Based on the inverse Fourier transformation of equation (

According to the definition, the input power flow of a submerged cylindrical shell can be written as

Substituting the radial displacement

For the purpose of facilitating the comparison, it is necessary to nondimensionalize the input power flow as

Only when

As a result, the additional-damping numerical integral method is employed in the integrating process to avoid singularity. This method is to introduce damping factor

The integral interval of equation (

Input vibration power flow curves are obtained for a shell made of steel immersed in viscous fluids with the amplitude of the circumferential-distributed line force _{0} = 1 N/m. The shell dimensions and material as well as some physical parameters of the fluid are given in Table

Physical parameter of the shell and fluid.

^{−2}) | _{s} (kg·m^{−3}) | _{f} (kg·m^{−3}) | |||||
---|---|---|---|---|---|---|---|

Shell | 1.92 × 10^{11} | 0.3 | 7800 | 2.54 | 0.02 | — | — |

Fluid | — | — | — | — | 1500 | 1000 |

In order to study the influence of shear and the expansive coefficients of viscosity _{b} on the characteristics of input vibration power flow respectively, the following three sets of data are selected and referred to as Case 1, Case 2, and Case 3 [

The values of coefficients of viscosity _{b}.

Coefficients of viscosity | Case 1 | Case 2 | Case 3 |
---|---|---|---|

^{−1}·s^{−1}) | 0.95 | 0.95 | 0.000894 |

_{b} (kg·m^{−1}·s^{−1}) | 0.95 | 0.0025 | 0.0025 |

In the previous vibroacoustic research, the acoustic medium is always considered as ideal fluid with fluid viscosity ignored. That is to say, that the shear coefficient of viscosity _{b} are eliminated and there are only longitudinal waves considered in the ideal fluid field. This means that the velocity continuity conditions on the boundary layer only exist in the radial direction and all the algebraic terms relating to transverse vector components _{b} = 0 into the equations above and take them as of that in ideal fluids.

When the two coefficients of viscosity _{b} are infinitely close to zero and the calculation results converge, the influence of fluid viscosity is small enough to be neglected, so the solutions can be as approximated by using the results for the ideal fluid. After several iterations (i.e., _{b} = 10^{−10} kg·m^{−1}·s^{−1}, _{b} = 10^{−12} kg·m^{−1}·s^{−1}, and _{b} = 10^{−15} kg·m^{−1}·s^{−1}), the results tend to converge, which demonstrates that the choice of _{b} = 10^{−15} kg·m^{−1}·s^{−1} is adequate for this case. Therefore, the calculation results of _{b} = 10^{−15} kg·m^{−1}·s^{−1} will be compared to the literature values in the ideal fluid in order to verify the accuracy of the proposed method and reliability of the numerical algorithm.

Figure _{b} = 10^{−15} kg·m^{−1}·s^{−1} and the literature values (LV) [

Comparison between the calculation results and the literature values (

The influence of fluid viscosity on input power flow of the coupled system can be demonstrated by the comparison between the calculation results of a cylindrical shell submerged in ideal fluids and that in viscous fluids under the action of the circumferential-distributional cosine-line force. Figure

Input power flow in ideal and viscous fluids. (a)

It is quite obvious that both the curves below have crest and share a similar variation trend. Moreover, the following conclusions can also be drawn:

When the nondimensional frequency Ω is within the range of 0 to 0.2, the viscous curves are initially lower than those in ideal fluids. Then, they tend to overlap with each other and the former even outstrip the latter while circumferential mode order

The curves reach their peak when Ω is between 0.2 and 0.7 regardless of the consideration of fluid viscosity. However, the values become larger with fluid viscosity taken into account, and the difference is particularly remarkable at the peak of the curves with the maximum differential rate coming to 38%. As previously mentioned, the fluid loads on axial and circumferential directions are ignored in ideal fluids, and only the radial loads on the shell are investigated. Thus, it can be seen that the input power flow of external force is promoted within this frequency band due to the consideration of transverse vibration and the dampening effect of fluid viscosity.

As Ω is greater than 0.7, the input power flow in a cylindrical shell submerged in ideal fluids is stronger than that in viscous fluids, and the maximum differential rate is around 10%. It is much smaller compared to the situation in low-frequency stage and illustrates that the influence of fluid viscosity mainly concentrates in the lower band. One probable cause for this phenomenon is that the damping action of fluid viscosity on the vibration response of the shell is mainly focused in middle-high frequency band and the auxoaction mentioned above is counteracted, even transcended by the energy loss gradually.

When studying the dispersion characteristics of the coupled system, a new definition of the characteristic frequency is introduced, which means the frequency that a propagating wave just occurs [

The input power flow curves of the coupled system are shown in Figure

Input power flow under different shear coefficients of viscosity. (a)

Figure

Input power flow under different expansive coefficients of viscosity. (a)

The values of the input power flow in the coupled system increase, especially significantly at the peak of the curves, with expansive coefficient of viscosity _{b} in the view of overall trends. In addition, the attenuation and propagation domains of the input power flow are alternate from the figures above. Moreover, the following phenomena can also be observed on the basis of different frequency bands:

When the dimensionless frequency Ω is less than 0.2, the input power flow curves of the coupled system almost coincide under two cases, which shows that the influence of _{b} in this frequency band is very weak.

When Ω is between 0.2 and 0.4, the curves of Case 2 are higher than those of Case 1. In addition, the former have crests while the latter have troughs.

When Ω is within the range from 0.4 to 1, the curves of Case 1 all rise sharply, reach the peak position, and then drop rapidly, while the ones of Case 2 fall fast from the peak of the previous stage. In this band, the former are much higher than the latter, and the maximum gap is about 98%.

When Ω is greater than 1, the changes in the curves tend to be stable under two cases. In this band, the curves of Case 1 are slightly higher than those of Case 2, and the difference between them is about 15%.

According to the Newtonian viscous fluid theory, the expansive coefficient of viscosity is determined by the volume strain. In addition, the causes of the above phenomena are also related to the dispersion characteristics of the coupled system:

The increase of the expansive coefficient of viscosity _{b} will reduce the value of the axial wave number of the coupled system, and the wave number is inversely proportional to the wave length and velocity. According to the definition, the input power flow of the coupled system is directly in proportion to the velocity, and, consequently, the values of the curves of Case 1 become larger overall, especially at the crests.

Greater expansive coefficient of viscosity _{b} results in the increase of the occurring frequency of propagation waves, so the dimensionless frequency corresponding to the peak values of the curves increases as well.

The motion form of some propagating waves changes with the expansive coefficient of viscosity _{b} and the circumferential mode number _{b} and

In this paper, the effect of fluid viscosity on the characteristics of vibration power flow in an infinite circular cylindrical shell immersed in a viscous acoustic medium is studied. Firstly, the expression of the input power flow of an infinite cylindrical shell immersed in viscous flow field under the action of the circumferential distribution line force is derived. Then, additional-damping numerical integral method and ten-point Gaussian integral method are employed to obtain the curves of input power flow curve in the coupled system. The following conclusions are drawn from the analysis:

The influence of fluid viscosity on the curves of the input power flow of the coupled system is mainly concentrated in the low-frequency band (0.2 < Ω < 0.7).

The input power flow of the coupled system increases in general after considering fluid viscosity, especially at the crests. Meanwhile, the number of the curve peaks increases, as well as the corresponding dimensionless frequencies.

Shear coefficient of viscosity

Greater increase of the expansive coefficient of viscosity _{b} results in a larger input power flow of the coupled system, especially at the crests. In addition, the dimensionless frequencies corresponding to the curve peaks increase while the attenuation and propagation domains change accordingly.

Based on the present work, the vibroacoustic characteristics of circular cylindrical shells with finite length, which is filled with a viscous acoustic medium, will be carried out considering the external dynamic loads and the hydrostatic pressure in the following work. Furthermore, the influence of fluid viscosity in the vibroacoustic coupled system will be revealed to improve the precision of noise source identification according to Nearfield Acoustic Holography (NAH) model.

Circular frequency

_{0}:

Amplitude of the circumferential-distributed line force

Displacements of shell in the

_{ns}

_{ns}

_{ns}:

Amplitudes of

Expression of

Circumferential mode order

Serial number of solutions of the dispersion equation in axial wave numbers

_{x}

_{θ}

_{r}:

Equivalent loads on the thin-walled cylindrical shell applied by the medium in the

Vibrational frequency

Nondimensional frequency of the elastic shell

Nondimensional axial wave number

Velocity components of the waves in

_{r}

_{θ}

_{r}:

Stress components of the waves in

Young’s modulus of the shell

Complex Young’s modulus of the shell

Poisson’s ratio of the shell

_{s}:

Material density of the shell

Shell’s mean radius

Thickness of the shell

_{f}:

Fluid density

_{f}:

Ideal speed of sound evaluated at ambient conditions

Sound pressure

_{f}:

Fluid velocity vector

Superposition of longitudinal and transverse vector components

Shear coefficient of viscosity

_{b}:

Expansive coefficient of viscosity

_{ns}:

Axial wave number

_{c}:

Complex compressible wave number

_{s}:

Complex shear wave number

Cylindrical Hankel’s functions of the second kind

_{ns}

_{ns}

_{ns}:

Unknown coefficients

_{ns}, Ψ

_{ns}:

Feature vectors

Force in complex form

Velocity in complex form

Period

_{input}:

Input power flow

Nondimensional input power flow

Damping factor.

No data were used to support this study.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The authors wish to express their gratitude to the Natural Science Foundation of Hainan Province of China (Grant no. 519QN188) and the National Natural Science Foundation of China (Grant no. 51709069) for their support.