非线性声谐振腔特性及气体动理学BGK格式模拟
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摘要
在声或热声装置的开发与应用中,大振幅声谐振腔已日益成为改善功率密度和热声转换效应的关键。然而,谐振腔中大振幅的声振荡,同时会引起诸多的非线性耗散,这也是应用者需要关注并着力解决的关键问题。因此,深刻认识谐振腔中各种非线性耗散机制并能抑制或减少各种非线性耗散以提高声幅的研究,对实际的应用至关重要。目前,人们对这些问题的认识和研究还远远不够。无论是采用理论分析、传统的模拟方法、还是实验方法开展研究,都有相当的挑战性,因而需要开拓新的研究思路。
本文的研究目的是应用开发一种适合研究谐振腔中非线性问题的有效数值算法,探索谐振管中各种非线性耗散的本质机理和重要的影响因素,寻找抑制或减少非线性耗散的有效方法,获得大幅声谐振腔的优良特性,满足实际的应用需要。考虑到气体动理学BGK (GKS-BGK)方法的优势特点,基于GKS-BGK方法开展了关于大幅声谐振腔的基本信息、特性规律、损失机理和设计理论的模拟研究,并辅以理论分析,对谐振腔内各种非线性耗散机理及其抑制方法进行了深入地探索。
本文的研究内容包括GKS-BGK模型的扩展、数值模拟和理论分析三个部分。主要创新性的学术贡献在于以下方面的内容。具体地,建模方面,采用高维可压缩粘性流体模型N-S (Navier-Stokes)方程,取代一维模型；同时,为模拟研究变截面谐振腔中非线性声振荡特性及二维流场,开发了非结构三角形网格上的GKS-BGK算法策略,并建立了相应的数学模型；数值模拟方面,采用GKS-BGK方法对等截面及变截面谐振腔内受迫运动气体的非线性声振荡特性及耗散机理进行了深入的模拟研究,并自主开发了模拟程序及相应的计算软件。理论分析方面,发展了一种适合变截面谐振管内非线性声振荡特性研究的一维理论分析方法。
本文主要研究内容概括如下。
(1)采用二维矩形网格上的GKS-BGK模型及算法模拟研究了等截面热声谐振管内有限振幅非线性声振荡,获得了与早期的理论分析、数值模拟、以及实验研究一致的结果,验证了模型算法的正确性。同时,研究了谐频附近七个不同驱动频率下声压演化过程、流场的瞬态和稳态分布以及谐频下二维流场中各种非线性效应的详细信息,特别分析了诸如激波、高次谐波、非线性振幅饱和、谐波耦合和基波到谐波能量级联过程等问题的机理。
(2)对等截面谐振腔内有限振幅和大振幅非线性声振荡的耗散和能量损失进行了分析。采用被验证了的二维GKS-BGK模型模拟研究了从线性、有限振幅到大振幅五个独立驱动幅值下的声场和流场特性,重点研究了大振幅振荡中的各种非线性效应,并分析了驱动位移幅值对压力、速度波形,瞬时流场和平均流场的影响,以及诸如激波、高次谐波、声流、涡流、边界层效应等非线性能量损失机制,为后面研究减少或抑制这些损失及提高声幅的相关措施积累数据基础。
(3)理论分析了变截面声谐振腔中非线性声传播的机理。基于理想气体状态下的控制方程,利用渐近展开得到几种不同形状谐振管的谐振模态频率表达式。然后重点针对指数型谐振管获得各物理量的近似解析表达式,为大振幅声谐振管的优化设计提供了理论基础,也为数值模拟提供了可供比较的验证和依据。
(4)构建了适合模拟变截面谐振腔中非线性效应的非结构三角形网格上的GKS-BCK模型,并采用该模型模拟研究了指数型谐振管内大振幅非线性声振荡特性和声流特性。首先揭示了指数型谐振管在抑制激波和高次谐波,提高基波幅值和压比方面的有效作用,并得到获得最大压比的最优管形参数,这个参数值和理论计算结果相吻合。然后对比研究了不同驱动下等截面谐振管和最优指数型管内的声场特性及声流场模式。这既是数值算法上的创新,也是对二维变截面大振幅声振荡这类国际学术前沿研究课题的推进。
With the development and application of acoustic or thermo-acoustic devices, the high amplitude acoustic resonator becomes a key component for increasing power density and improving thermoacoustic effects. Yet, the high amplitude acoustic oscillations in the resonator will always cause much undesired nonlinear dissipation, which is a dilemma for researchers and engineers to face and necessarily solve in thermoacoustic applications. Therefore, it is significantly of importance for researchers to have a good knowledge of the mechanisms and main factors inducing various nonlinear dissipations, as well as of some methods to suppress or decrease these dissipations in actual thermoacoustic applications. So far, some nonlinear phenomena and problems have not yet been well understood and solved, because of the relatively big challenges in theory analysis, traditional numerical modeling as well as experiment measurements. Thus new approaches are needed to explore, which is the motivation of this dissertation.
The research objection of this thesis is to apply and develop an effective numerical algorithm suitable for modeling nonlinear problems in an acoustic or thermoacoustic resonator, to explore the intrinsic mechanisms and main factors inducing various nonlinear dissipation, to search effective measures for suppressing or decreasing these dissipation, as well as to obtain the good characteristic of a high amplitude acoustic resonator for practice applications. For this purpose, taking into account some advantages of the gas-kinetic scheme based on BGK (GKS-BGK), this thesis conducts some numerical simulations through GKS-BGK for acquiring relevant basic information, regular characteristic, mechanism of loss, as well as design theory about high amplitude acoustic resonators. In addition, the mechanism of nonlinear dissipation and relevant effective methods to suppress them are closely studied and analyzed numerically and theoretically.
The content of this thesis is composed of three parts, including the extended GKS-BGK model, numerical simulation and theory analysis. The main creative academic contributions are in that the following aspects. Specifically, in establishing of a model, the multi-dimension N-S (Navier-Stokes) equations instead of the one-dimension model are used to describe the flow motions in the resonator. Moreover, a GKS-BGK on the unstructured triangle mesh is constructed to simulate the nonlinear oscillations and two- dimension flow fields in the resonator with variable cross-section, and relevant model is established; Numerically, these characteristics of nonlinear oscillations and the dissipation mechanisms are closely investigated by the developed GKS-BGK for both the constant and the variable cross sectional resonant tube. Moreover a self-owned program code and the simulation software are developed. Theoretically, a one dimensional analytic method is proposed for predicting the characteristics of nonlinear oscillations in a variable cross-section resonator.
The following summarizes the main studied work of this thesis.
(1) Nonlinear oscillations in a resonator with constant cross-section are simulated by the GKS-BGK based on a two-dimensional rectangle mesh. Simulated results are obtained in consistence with those of previous research from theoretical, numerical and experimental literature. And the model algorithm is thus verified. Additionally, the evolution process of acoustic pressure and transient as well as steady-states flow fields are closely investigated taking seven separate driving frequencies near the resonant frequence into accounts. Also obtained is the detailed information on various nonlinear effects in two-dimensional flow fields at resonance. Of these nonlinear effects, shock waves, high order harmonic waves, acoustic saturation, coupling of harmonic waves, as well as energy cascade process from fundamental wave to harmonic waves are especially analyzed for searching their nonlinear dissipation mechanisms.
(2) For finite amplitudes and large amplitudes, the analyses are carried out on the nonlinear dissipation and energy losses in a constant cross section resonator. The characteristics of acoustic fields and flow fields are studied by the previously verified two-dimensional GKS-BGK model for five separate driving displacement amplitudes ranging from linear to high amplitude domain. Nonlinear effects in high amplitude of acoustic oscillations are closely investigated. The effect of the driving displacement amplitude is in detail numerically analyzed on waveforms of pressure and velocity waves, transient steady-state flow fields as well as average mass flow fields. Additionally, the mechanisms of nonlinear energy losses, such as shock wave, higher harmonic, acoustic streaming, vortex, as well as boundary layer effect are also explored. These researches above will provide some basic numerical data for enhancing acoustic amplitude by some measures to decrease or suppress these undesired losses.
(3) The propagating mechanism of nonlinear acoustics in a varying cross-section resonator is analyzed theoretically. These frequence relations between resonant modes for several different shaped resonators are independently derived and obtained through the asymptotic expansion method for ideal gas. Especially, the approximate analytic solution is obtained for acoustic variables in an exponential resonator. These theoretically analytic results are significantly helpful for the optimal design of high amplitude acoustic resonators, as well as for a verification and comparison of numerical results.
(4) A GKS-BGK model based on the unstructured triangle mesh is constructed for simulating nonlinear effects in the variable cross-section resonator. And, the characteristics of high amplitude acoustic oscillations and acoustic streaming in an exponential resonator are simulated using this self-designed GKS-BGK model. First, the effective function of an exponential resonator is revealed in suppressing the shock wave and higher harmonic, as well as in enhancing acoustic amplitude and pressure ratio. An optimal flare parameter value is obtained for an exponential resonator and is in consistence with the theoretical one. Then, the changes of the acoustic field characteristic and streaming pattern with the driving amplitude are compared between the constant and the varying cross-section resonator. These model algorithms belong to author's creative work, which will provide a new approach and prompt the advance in the research of varying cross-section resonator.
本文的研究目的是应用开发一种适合研究谐振腔中非线性问题的有效数值算法,探索谐振管中各种非线性耗散的本质机理和重要的影响因素,寻找抑制或减少非线性耗散的有效方法,获得大幅声谐振腔的优良特性,满足实际的应用需要。考虑到气体动理学BGK (GKS-BGK)方法的优势特点,基于GKS-BGK方法开展了关于大幅声谐振腔的基本信息、特性规律、损失机理和设计理论的模拟研究,并辅以理论分析,对谐振腔内各种非线性耗散机理及其抑制方法进行了深入地探索。
本文的研究内容包括GKS-BGK模型的扩展、数值模拟和理论分析三个部分。主要创新性的学术贡献在于以下方面的内容。具体地,建模方面,采用高维可压缩粘性流体模型N-S (Navier-Stokes)方程,取代一维模型；同时,为模拟研究变截面谐振腔中非线性声振荡特性及二维流场,开发了非结构三角形网格上的GKS-BGK算法策略,并建立了相应的数学模型；数值模拟方面,采用GKS-BGK方法对等截面及变截面谐振腔内受迫运动气体的非线性声振荡特性及耗散机理进行了深入的模拟研究,并自主开发了模拟程序及相应的计算软件。理论分析方面,发展了一种适合变截面谐振管内非线性声振荡特性研究的一维理论分析方法。
本文主要研究内容概括如下。
(1)采用二维矩形网格上的GKS-BGK模型及算法模拟研究了等截面热声谐振管内有限振幅非线性声振荡,获得了与早期的理论分析、数值模拟、以及实验研究一致的结果,验证了模型算法的正确性。同时,研究了谐频附近七个不同驱动频率下声压演化过程、流场的瞬态和稳态分布以及谐频下二维流场中各种非线性效应的详细信息,特别分析了诸如激波、高次谐波、非线性振幅饱和、谐波耦合和基波到谐波能量级联过程等问题的机理。
(2)对等截面谐振腔内有限振幅和大振幅非线性声振荡的耗散和能量损失进行了分析。采用被验证了的二维GKS-BGK模型模拟研究了从线性、有限振幅到大振幅五个独立驱动幅值下的声场和流场特性,重点研究了大振幅振荡中的各种非线性效应,并分析了驱动位移幅值对压力、速度波形,瞬时流场和平均流场的影响,以及诸如激波、高次谐波、声流、涡流、边界层效应等非线性能量损失机制,为后面研究减少或抑制这些损失及提高声幅的相关措施积累数据基础。
(3)理论分析了变截面声谐振腔中非线性声传播的机理。基于理想气体状态下的控制方程,利用渐近展开得到几种不同形状谐振管的谐振模态频率表达式。然后重点针对指数型谐振管获得各物理量的近似解析表达式,为大振幅声谐振管的优化设计提供了理论基础,也为数值模拟提供了可供比较的验证和依据。
(4)构建了适合模拟变截面谐振腔中非线性效应的非结构三角形网格上的GKS-BCK模型,并采用该模型模拟研究了指数型谐振管内大振幅非线性声振荡特性和声流特性。首先揭示了指数型谐振管在抑制激波和高次谐波,提高基波幅值和压比方面的有效作用,并得到获得最大压比的最优管形参数,这个参数值和理论计算结果相吻合。然后对比研究了不同驱动下等截面谐振管和最优指数型管内的声场特性及声流场模式。这既是数值算法上的创新,也是对二维变截面大振幅声振荡这类国际学术前沿研究课题的推进。
With the development and application of acoustic or thermo-acoustic devices, the high amplitude acoustic resonator becomes a key component for increasing power density and improving thermoacoustic effects. Yet, the high amplitude acoustic oscillations in the resonator will always cause much undesired nonlinear dissipation, which is a dilemma for researchers and engineers to face and necessarily solve in thermoacoustic applications. Therefore, it is significantly of importance for researchers to have a good knowledge of the mechanisms and main factors inducing various nonlinear dissipations, as well as of some methods to suppress or decrease these dissipations in actual thermoacoustic applications. So far, some nonlinear phenomena and problems have not yet been well understood and solved, because of the relatively big challenges in theory analysis, traditional numerical modeling as well as experiment measurements. Thus new approaches are needed to explore, which is the motivation of this dissertation.
The research objection of this thesis is to apply and develop an effective numerical algorithm suitable for modeling nonlinear problems in an acoustic or thermoacoustic resonator, to explore the intrinsic mechanisms and main factors inducing various nonlinear dissipation, to search effective measures for suppressing or decreasing these dissipation, as well as to obtain the good characteristic of a high amplitude acoustic resonator for practice applications. For this purpose, taking into account some advantages of the gas-kinetic scheme based on BGK (GKS-BGK), this thesis conducts some numerical simulations through GKS-BGK for acquiring relevant basic information, regular characteristic, mechanism of loss, as well as design theory about high amplitude acoustic resonators. In addition, the mechanism of nonlinear dissipation and relevant effective methods to suppress them are closely studied and analyzed numerically and theoretically.
The content of this thesis is composed of three parts, including the extended GKS-BGK model, numerical simulation and theory analysis. The main creative academic contributions are in that the following aspects. Specifically, in establishing of a model, the multi-dimension N-S (Navier-Stokes) equations instead of the one-dimension model are used to describe the flow motions in the resonator. Moreover, a GKS-BGK on the unstructured triangle mesh is constructed to simulate the nonlinear oscillations and two- dimension flow fields in the resonator with variable cross-section, and relevant model is established; Numerically, these characteristics of nonlinear oscillations and the dissipation mechanisms are closely investigated by the developed GKS-BGK for both the constant and the variable cross sectional resonant tube. Moreover a self-owned program code and the simulation software are developed. Theoretically, a one dimensional analytic method is proposed for predicting the characteristics of nonlinear oscillations in a variable cross-section resonator.
The following summarizes the main studied work of this thesis.
(1) Nonlinear oscillations in a resonator with constant cross-section are simulated by the GKS-BGK based on a two-dimensional rectangle mesh. Simulated results are obtained in consistence with those of previous research from theoretical, numerical and experimental literature. And the model algorithm is thus verified. Additionally, the evolution process of acoustic pressure and transient as well as steady-states flow fields are closely investigated taking seven separate driving frequencies near the resonant frequence into accounts. Also obtained is the detailed information on various nonlinear effects in two-dimensional flow fields at resonance. Of these nonlinear effects, shock waves, high order harmonic waves, acoustic saturation, coupling of harmonic waves, as well as energy cascade process from fundamental wave to harmonic waves are especially analyzed for searching their nonlinear dissipation mechanisms.
(2) For finite amplitudes and large amplitudes, the analyses are carried out on the nonlinear dissipation and energy losses in a constant cross section resonator. The characteristics of acoustic fields and flow fields are studied by the previously verified two-dimensional GKS-BGK model for five separate driving displacement amplitudes ranging from linear to high amplitude domain. Nonlinear effects in high amplitude of acoustic oscillations are closely investigated. The effect of the driving displacement amplitude is in detail numerically analyzed on waveforms of pressure and velocity waves, transient steady-state flow fields as well as average mass flow fields. Additionally, the mechanisms of nonlinear energy losses, such as shock wave, higher harmonic, acoustic streaming, vortex, as well as boundary layer effect are also explored. These researches above will provide some basic numerical data for enhancing acoustic amplitude by some measures to decrease or suppress these undesired losses.
(3) The propagating mechanism of nonlinear acoustics in a varying cross-section resonator is analyzed theoretically. These frequence relations between resonant modes for several different shaped resonators are independently derived and obtained through the asymptotic expansion method for ideal gas. Especially, the approximate analytic solution is obtained for acoustic variables in an exponential resonator. These theoretically analytic results are significantly helpful for the optimal design of high amplitude acoustic resonators, as well as for a verification and comparison of numerical results.
(4) A GKS-BGK model based on the unstructured triangle mesh is constructed for simulating nonlinear effects in the variable cross-section resonator. And, the characteristics of high amplitude acoustic oscillations and acoustic streaming in an exponential resonator are simulated using this self-designed GKS-BGK model. First, the effective function of an exponential resonator is revealed in suppressing the shock wave and higher harmonic, as well as in enhancing acoustic amplitude and pressure ratio. An optimal flare parameter value is obtained for an exponential resonator and is in consistence with the theoretical one. Then, the changes of the acoustic field characteristic and streaming pattern with the driving amplitude are compared between the constant and the varying cross-section resonator. These model algorithms belong to author's creative work, which will provide a new approach and prompt the advance in the research of varying cross-section resonator.
引文
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