Sound generation and transmission in flow ducts with axial temperature gradients

This article describes a one-dimensional, linearized, analysis of fundamental mode sound generation and propagation in rigid-walled flow ducts with axial temperature variation. An acoustic wave equation, including damping effects and volume sources, is derived and its solution (in the absence of sources) by a numerical technique and an approximate analytical method is discussed. The “forced” wave equation is then solved (the existence of an oscillating solution to the “unforced” equation being assumed) for sound generation by a side-branch volume source in an infinite duct, and the results are applied to a duct of finite length. Reasonably good agreement is obtained between measurements and predictions of the sound pressure field in a flow duct, away from the source region.     

On the one-dimensional acoustic propagation in conical ducts with stationary mean flow

This paper proposes a direct time-domain calculation of the time-domain responses of anechoic conical tubes with steady weak mean flow. The starting point is the approximated linear one-dimensional wave equation governing the velocity potential for the case of steady flow with low Mach number. A traveling solution with general space-dependent propagation velocity is then proposed from which the inward and outward pressure and velocity impulse responses can be obtained. The results include the well-known responses of conical and cylindrical ducts with zero mean flow.     

Effect of flow on quasi-one-dimensional acoustic wave propagation in a variable area duct of finite length

The general equation for the velocity potential of quasi-one-dimensional acoustic wave motion in a variable area, finite duct with one-dimensional flow is derived by using a perturbation technique. The non-linear second-order partial differential equation is linearized and then solved, by either a power series expansion method or the Runge-Kutta fourth-order method, for harmonic time dependence. The boundary condition taken at the duct mouth is that of matching the impedance of the duct sound field to that of the radiation field at the duct opening. Three axial Mach number variations along the duct axis are considered and the results obtained are compared with those for the case of constant Mach number, to determine the influence of the axial velocity gradient on sound propagation. The effect of flow on the radiation impedance is also considered.     

Application of dual reciprocity boundary element method for prediction of acoustic characteristics of ducts and silencers with three-dimensional potential flow

The Dual Reciprocity Boundary Element Method （DRBEM） is applied to predict the acoustic characteristics of ducts and silencers with three-dimensional potential flow, and the basic principle and numerical procedure of the proposed method are introduced. Compared to the Conventional Boundary Element Method （CBEM）, the DRBEM takes into account the second order terms of flow Mach number in the acoustic governing equation, which is suitable for the situations with higher Mach number subsonic flow. The four-pole parameters of a duct and a varying cross-sectional area expansion chamber are predicted with the DRBEM, and the predictions are compared with the one-dimensional analytical solutions and the CBEM results. The comparisons demonstrated that the present method is valid. Transmission loss of silencers with different structures was also calculated with the DRBEM. The results showed that the influence of the three-dimensional flow on the acoustic characteristics of silencers with complex structures is not negligible.     

Cyclostrophic adjustment in swirling gas flows and the Ranque-Hilsch vortex tube effect

A theoretical analysis of cyclostrophic adjustment is presented; i.e., adjustment to balance between pressure gradient and centrifugal force in axisymmetric flow of an inviscid gas is examined. The solution to the problem is represented as the sum of a time-independent (balanced) and time-dependent (wave) components. It is shown that the wave component of the flow in an unbounded domain decays with time, and the corresponding solution reduces to the balanced component. In a bounded domain, the balanced flow component exists against the background of undamped acoustic waves. It is found that the balanced flow is thermally stratified at Mach numbers close to unity, with a substantial decrease in gas temperature (to between ?50 and ?100°C) in the axial region. This finding, combined with the results of special experiments, is used to explain the Ranque-Hilsch vortex tube effect.     

FINITE ELEMENT AND BOUNDARY ELEMENT MODELLING FOR THE ACOUSTIC WAVE TRANSMISSION IN MEAN FLOW MEDIUM

The frequency characteristics of the acoustic wave transmission in a medium with mean flow are considered. One approach is to solve the Helmholtz equation with mean flow medium in original co-ordinates, which is directly discretized for the one-dimensional and the axisymmetric FEM. Another approach is to transform the equation into the standard Helmholtz equation, which is discretized for the axisymmetric FEM and the three-dimensional BEM. The numerical models are examined first for a straight circular duct. The solutions by the numerical approaches are compared with the analytical solution. The examination is then extended to the case when the mean flow is locally present in the muffler with expansion chamber. To model the spatial mean flow in the BEM model, the partitioned domain approach is also developed. No shear effect between the two regions are included.     

Exact solutions for modeling sound propagation through a combustion zone.

Exact analytical solutions for one-dimensional sound propagation through a combustion zone, taking the effects of mean temperature gradient and oscillatory heat release into account, are presented in this paper. The wave equation is derived starting from the momentum and energy equations. Using appropriate transformations, solutions are derived for the case of an exponential mean temperature gradient in terms of Bessel functions. For the case of a linear mean temperature profile, solutions are derived in terms of confluent hypergeometric functions. Example calculations show that the accuracy in modeling combustion-acoustics interactions can be significantly increased by the use of these solutions.     

ON TRANSMISSION OF SOUND IN A NON-UNIFORM DUCT CARRYING A SUBSONIC COMPRESSIBLE FLOW

The differential equations governing the transmission of one-dimensional sound waves in a non-uniform duct carrying a subsonic compressible mean flow have been the subject of a recent debate [1, 2]. Of the two formulations presented, one is considered to be non-acoustical and the other as neglecting the spatial variation of the speed of sound. The present paper shows that both formulations are acoustical and represent valid approximations to correct conditions for isentropic sound propagation in a subsonic low Mach number duct. Each formulation is associated with an “error wave”, which is essentially a hydrodynamic wave when the mean flow Mach number is small. Three-port modelling is required, however, to capture this wave when the Mach number of the mean flow is relatively large and a numerical matrizant approach is described which can be used for this purpose.     

除垢超声波传播影响因素的理论研究

从一维平面波理论入手分析了超声波声压分布特性。依据多普勒频移原理,在声场的运动方程,连续性方程,波动方程的基础上,建立一个超声波在流动的液体中传播的控制方程。根据轴对称模型的实际特点,简化了所得方程,并求出解析解。结果表明流动液体可以产生声波的衰减。液体的黏滞性是产生超声波衰减的重要原因。超声波的频率较高,液体的黏滞性对超声波衰减影响明显。依据黏滞力与速度梯度的关系,建立一个超声波在黏滞液体传播的控制方程,并依据边界条件求出解析解,反映了媒质黏滞性对超声波传播尤其是衰减特性的影响。     

低马赫数方法在自然对流数值模拟的应用

采用低马赫数近似的方法来对大温差驱动的自然对流问题进行数值模拟。低马赫数近似是通过将全可压的Navier—Stokes方程中声波进行过滤,从而在马赫数较低的流动中忽虑声波对流场的影响。声波过滤后的方程具有不可压缩N—S方程的特点,但可以求解温度和密度变化较大的问题。首先,通过对盖顶驱动流的数值模拟,验证了本文方法的可靠...     

Application of stochastic computer simulation to a solution of a nonlinear Fokker-Planck equation governing bacteria chemotaxis

A stochastic computer simulation technique has been used to solve a one-dimensional time-dependent Fokker-Planck (F-P) equation governing spacetime distribution of bacteria in a substrate (food) gradient. Because of consumption of the substrate by the bacteria the F-P equation becomes nonlinear due to coupling of the distribution with the self-generated substrate gradient. The simulation is efficient and numerically accurate for generating transient solutions. We are able not only to produce a transient solution displaying approach to the steady state solitary wave solution known from an analytical result of Keller and Segal but also to show an interesting dependence of the solitary wave propagation speed on the concentration dependence of the substrate consumption rate.     

On the effect of viscosity and thermal conductivity on sound propagation in ducts: A re-visit to the classical theory with extensions for higher order modes and presence of mean flow

The dispersion equation for the axisymmetric modes of viscothermal acoustic wave propagation in uniform hard-walled circular ducts containing a quiescent perfect gas is classical. This has been extended to cover the non-axisymmetric modes and real fluids in contemporary studies. The fundamental axisymmetric mode has been the subject of a large number of studies proposing approximate solutions and the characteristics of the propagation constants for narrow and wide ducts with or without mean flow is well understood. In contrast, there are only few publications on the higher order modes and the current knowledge about their propagation characteristics is rather poor. On the other hand, there is a void of papers in the literature on the effect of the mean flow on the quiescent modes of propagation. The present paper aims to contribute to the filling of these gaps to some extent. The classical theory is re-considered with a view to cover all modes of acoustic propagation in circular ducts carrying a real fluid moving axially with a uniform subsonic velocity. The analysis reveals a new branch of propagation constants for the axisymmetric modes, which appears to have escaped attention hitherto. The solution of the governing wave equation is expressed in a modal transfer matrix form in frequency domain and numerical results are presented to show the effects over wide ranges of frequency, viscosity and mean flow parameters on the propagation constants. The theoretical formulation allows for the duct walls to have finite impedance, but no numerical results are presented for lined ducts or ducts carrying a sheared mean flow.     

Non-linear peristaltic transport of a Newtonian fluid in an inclined asymmetric channel through a porous medium

Peristaltic transport of an incompressible viscous fluid in an inclined asymmetric channel through a porous medium is studied under long-wavelength and low-Reynolds number assumptions. The flow is examined in a wave frame of reference moving with the velocity of the wave. The analytical solution has been obtained in the form of a stream function from which the axial velocity and pressure gradient have been derived. The results for the pressure drop and shear stress have also been computed numerically. The effects of various physical parameters are discussed through graphs and the phenomenon of trapping is also discussed. Comparison of various wave forms (namely sinusoidal, triangular, square and trapezoidal) on the flow is discussed.     

An integral equation for radiation transport in an infinite cylinder with Fresnel reflection

We formulate an integral equation for radiation transport in an infinitely long cylinder. Scattering is included and sources arise from incidence on the surface and from an internal volume source. Internal reflection at the surface follows the Fresnel law. For the special case of no scattering and no axial variation of the source, we obtain an exact solution which we have compared numerically with some results of Tian and Chiu [Radiative absorption in an infinitely long hollow cylinder with Fresnel surfaces. JQSRT 2006;98:249]. We have also obtained closed form analytical solutions for no scattering with a line source along the cylinder axis and also that for the case of a spatially constant volume source. The general case when there is variation in the axial direction is also presented and the special case of a radially uniform plane source at the origin is explored in some detail. The axial solution is compared with an approximate areal average method introduced by Larsen [A one-dimensional model for three dimensional transport in a pipe. Transp Theory Stat Phys 1984;13:599; Larsen EW, Malvagi F, Pomraning GC. One dimensional models for neutral particle transport in ducts. Nucl Sci Eng 1986;93:13] in another context.     

管道后传声数值模拟的不稳定波抑制

在管道后传声的数值模拟中,必须考虑平均流剪切层的散射效应,然而在非均匀剪切流动下时域求解线化欧拉方程会面临Kelvin-Helmholtz不稳定波产生和放大的难题。已有的不稳定波抑制技术通常很难获得令人满意的结果。本文采用一种混合方法,首先引入有限时段的宽频声源波包将声波和不稳定波分离,进而采用声源滤波器技术对不稳定波进行抑制。数值验证算例选择半无限长轴对称环形硬壁直管道,采用计算气动声学方法时域求解2.5维线化欧拉方程,无背景流动的数值解与解析解符合很好,验证了程序的精度与可靠性,非均匀流动算例则表明所采用波包加声源滤波器混合方法对不稳定波抑制效果明显,对声场影响很小,充分显示了该方法的精度与可行性。     

Acoustic-Gravity Waves in the Atmosphere with a Piecewise-Linear Temperature Profile

We consider the problem of propagation of acoustic-gravity waves in the atmosphere with a constant temperature gradient in the near-surface layer. The assumption of linear temperature dependence on height allowed us to reduce the wave equation to the hypergeometric form, regardless of the compressibility of the medium. The solution of this equation is represented in terms of degenerate hypergeometric functions. To analyze the obtained solution, we consider a two-layer model of a half-bounded atmosphere with a height-independent background temperature in the upper layer. The results are studied in detail under the approximation of an incompressible medium. For the model specified above, we find analytical expressions for the perturbation fields and obtain a characteristic equation whose solution allows us to calculate wave dispersion characteristics at frequencies close to the Brunt-Väisälä frequency for large horizontal scales as compared to the layer thickness.     

Numerical study on the gas-kinetic high-order schemes for solving Boltzmann model equation

The high-order compact finite difference technique is introduced to solve the Boltzmann model equation, and the gas-kinetic high-order schemes are developed to simulate the different kinetic model equations such as the BGK model, the Shakhov model and the Ellipsoidal Statistical (ES) model in this paper. The methods are tested for the one-dimensional unsteady shock-tube problems with various Knudsen numbers, the inner flows of normal shock wave for different Mach numbers, and the two-dimensional flows past ...     

Sound Attenuation in a Circular Duct of a Viscous Medium in the Absence of Mean Flow

Abstract

An analytical method to study the effect of viscosity of a medium and the wave number on sound propagation and sound attenuation numbers in circular ducts has been presented. The method is based on the variation of parameters of the solution corresponding to the case of inviscid acoustic waves in circular ducts and axisymmetric modes. A mathematical model is constructed to describe the physical problem in general. Three basic assumptions have been considered, namely, each flow quantity has been written as the sum of a steady mean flow and an unsteady acoustic flow quantity. The effect of thermal conductivity of the gas has been neglected as well as no mean flow. The results for a wide range of wave numbers and Reynolds numbers show that for a viscous medium, the propagation number is a weak function of the Reynolds number, and as the Reynolds number increases, the propagation number approaches its inviscid value. Also the propagation number is independent of the wave number. For the attenuation number, it decreases monotonically with the increase of the Reynolds number and it vanishes when Reynolds number exceeds 10⁴.     

Thermoelastic damping in thin microrings with two-dimensional heat conduction

Accurate determination of thermoelastic damping (TED) is very challenging in the design of micro-resonators. Microrings are widely used in many micro-resonators. In the past, to model the TED effect on the microrings, some analytical models have been developed. However, in the previous works, the heat conduction within the microring is modeled by using the one-dimensional approach. The governing equation for heat conduction is solved only for the one-dimensional heat conduction along the radial thickness of the microring. This paper presents a simple analytical model for TED in microrings. The two-dimensional heat conduction over the thermoelastic temperature gradients along the radial thickness and the circumferential direction are considered in the present model. A two-dimensional heat conduction equation is developed. The solution of the equation is represented by the product of an assumed sine series along the radial thickness and an assumed trigonometric series along the circumferential direction. The analytical results obtained by the present 2-D model show a good agreement with the numerical (FEM) results. The limitations of the previous 1-D model are assessed.     

Acoustic wave motion along a narrow cylindrical duct in the presence of an axial mean flow and temperature gradient

A numerical solution is presented to the problem of nonisentropic acoustic wave motion in a circular capillary tube in the presence of both axial mean flow and a background axial temperature gradient. The effects of the radial components of the acoustic velocity are included in the analysis. The main application area is in the study of the acoustic effects of catalytic converters. The solution makes use of a series expansion and is valid for small relative changes in the background temperature, which are typical of this application area. Various solutions to the problem have been obtained previously, using different simplifications to the complete problem which is considered here. It is shown that each of the simplifications results in errors for the predicted attenuation of at least 20 dB/m, using conditions typical for catalytic converters. In particular, the isentropic assumption is shown to be invalid.