Aeroacoustics_气动声学(理论教程)

Aeroacoustics 气动声学
翻译：岳刚伟
简介
本翻译英文原文源于STAR-CCM+12.02版本的帮助文件，仅供从事CFD相关领域的同学参考，译者从2010年开始从事汽车行业的CFD仿真分析工作，本翻译根据自身的理解进行，翻译过程中错误在所难免，请予以指正。

附制作的空气动力学视频，请提出指导建议，感谢！
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Computational aeroacoustics (CAA) is a branch of multiphysics modeling and simulation that involves identifying noise sources that are induced by fluid flow and propagation of the subsequently generated sound waves.
计算气动声学（CAA）是多体物理学的建模和仿真的一个分支，包括识别流体流动和随后产生的声波的传递而产生的噪声源。

Noise sources originate from various types of flow, such as:
噪声源来自于各种类型的流动，例如：
Turbulent flow over solid bodies (bluff body flows)
固体表面的湍流（钝体/非线性流动）
Turbulent boundary layer flows (for example, automobile, aircraft components)
湍流边界层流动（例如汽车、飞机部件）
High-speed turbulent shear flows (for example, free jet flow) 高速湍流切变流动（例如，自由射流）
High-speed impinging flows (for example, jet impingement, rocket exhaust noise)
高速撞击流（如射流冲击、火箭排气噪声）
Structural vibration that is induced by fluid flow (fluid-structure interactions)
由流体流动（流体与结构相互作用）引起的结构振动
High-speed rotating flows (for example, rotorcrafts or turbomachinery)
高速旋转流（例如，直升机或涡轮机械）
Turbulent combustion (reacting flows)
湍流燃烧（反应流）
Blast waves (explosions)
爆炸波（爆炸）
A typical CAA simulation requires the following components:
典型的CAA仿真需要以下组件：
Navier-Stokes equations for fluid flow
流体流动的纳维-斯托克方程
High-resolution turbulence models
高精度的湍流模型
Analytical or computational acoustic wave propagation
models
解析或计算声波传播模型
The noise signatures at the locations of interest exhibit corresponding noise spectra—that is, the intensities of sound pressure level over a range of frequencies. The noise characteristic can be tonal noise with a distinct peak at a frequency (such as engine noise; jet impingement noise, or Noise, Vibration, and Harshness (NVH)) or broadband noise spread over a frequency range (typical of turbulence-induced noise).
关注部位的噪声特征表现出相应的噪声谱，即声压级在某一频率范围内的强度。

噪声的特性可以是离散的，在同一频率有不同的噪声峰值（如发动机噪声；射流冲击噪声，或则噪声，振动，和舒适性（NVH））或一定频率范围内传播的宽频噪声（典型的湍流引起的噪声）。

The sound pressure level (SPL measured in decibel, dB) is:
声压级(SPL测量，分贝)：
(3507)
where:
而：
root mean square pressure
压力均方根
reference pressure (usually 20 µPa)
参考压力（通常20µPa）
The mechanism of noise generation differs according to the underlying physics of the flow. Thus, a computational aeroacoustics method must focus on resolving noise generation by optimizing the mesh and solver settings to capture the noise sources and frequence relevant to acoustic analysis. Resolution of the noise sources relies on the fidelity of the turbulence modeling.
噪声产生的机理根据不同的流动的基础物理模型而不同。

因此，计算气动声学方法必须着眼于通过优化网格和求解器的设置来解决噪声的生成，并获取气动声学分析相关的噪声源和频率。

噪声源的精度依赖于湍流模型的准确性。

Although LES and DES simulations can predict noise generation in the near-field, they are not generally suitable for predicting noise propagation to far-field locations. The magnitudes of flow
fluctuations that generate noise are orders of magnitude lower than the hydrodynamic flow properties. Without adequate mesh resolution, sound waves quickly dissipate away from the source region. The mesh density that is required to preserve this acoustic signature at far distances restricts the feasibility of pure CFD for real world applications.
尽管LES和DES模拟可以预测近场产生的噪声，他们通常不适合预测噪声传播到远场的位置。

流动波动产生的噪声值的数量级低于流体流动的属性。

没有足够的网格精度，声波很快从声源区耗散。

对网格密度的要求是保证远距离的声学特征在纯CFD方法对现实世界的应用的可行性不被限定。

It is more practical to consider noise generation and propagation phenomena separately, adopting an appropriate hybrid CFD/CAA method. In this method, LES or DES is coupled (one-way) with a noise propagation aeroacoustic model.
考虑噪声的产生和传播现象，采用适当的混合CFD/CAA方法更为实用。

在这种方法中，LES和DES与噪声传播的声学模型进行耦合（单向）。

Noise propagation modeling dates back to the 1950s. Sir James Lighthill [790] formulated Eqn. (3554) based on the Navier-Stokes equations and some basic assumptions, containing the definition of quadrupole noise sources from turbulent flows.
噪声传播模型可以追溯到20世纪50年代。

Sir James Lighthill 基于纳维斯托克和一些基本的假定，包括湍流中四极子噪声源的定义，而建立的方程。

Following Lighthill's equation, other noise propagation models evolved considering all fundamental components of noise sources (monopole, dipole, and quadrupole), such as Curle's equation and the Ffowcs Williams-Hawkings equation (FWH). The surface integral methods such as those of the FWH method are a practical approach for noise propagation in the far-field, being able to superimpose all sources of noise over an enclosed surface and provide acoustic signatures by accurate meshless analytical solutions of a surface integral equation.
下面的Lighthill方程，其他衍生的噪声的传播模型是考虑噪声源的所有基本组成（单极子，偶极子和四极子），如Curle方程和Ffowcs Williams Hawkings 方程（FWH）。

曲面积分的方法如的FWH方法是在远场噪声传播的一个实用的方法，能够把所有的噪声源叠加在一个封闭的表面，并通过曲面积分方程的无网格化分析结果来提供准确的声学特征。

Other forms of acoustic wave equations can also be formulated, depending on the starting model equations and source definitions.
High fidelity LES/DES can be combined with other accurate mesh-based CAA methods (typically requiring high-order space and time discretization) such as Linearized Euler Equations (LEE) or Acoustic Perturbation Equations (APE), but not without extreme computational burden.
其他形式的声波方程也可以根据初始模型方程和源定义来制定。

高精度的LES / DES可以与其他精确的基于网格的CAA（计算气动声学）方法（通常需要高阶空间和时间离散化）相结合，例如线性化欧拉方程（LEE）或声学扰动方程（APE），但是需要极大的计算量。

STAR-CCM+ uses LEE and Curle for noise source modeling only, not for acoustic wave modeling (LEE), or for propagation (Curle).
The FWH method is based on the assumption of sound propagation in free space, so it cannot consider any embedded structure in the propagation region. On the other hand, a Boundary Element Method (BEM) for the solution of the Helmholtz equation (a wave equation in the frequency domain) by a boundary integral equation [references of BEM] can provide solutions of acoustic field with embedded structures.
STAR-CCM+ 单使用LEE和Curle进行噪声源建模，不是声波建模（LEE），或传播（Curle）。

FWH方法是基于声音在自由空间传播的假设，所以它不考虑任何深入结构内部的传播区域。

另一方面，利用边界积分方程（参考BEM）求解亥姆霍兹方程（频域中的波动方程）的边界元法（BEM）可以提供深入结构内部的声学解。

The BEM, however, is applicable only for low frequency ranges due to its computational expense. The hybrid method can be extended to Finite Element Analysis (FEA) or Statistical Energy Analysis (SEA) for high frequency applications.
但是，BEM，由于其计算费用，只适用于低频范围。

这种混合方法可以扩展到有限元分析(FEA)或统计能量分析（SEA）的高频应用。

In summary, the broad categories of computational aeroacoustics methods, based on feasibility, are:
总之，基于可行性计算气动声学方法的类，是
Prediction of near-field noise by Large-Eddy Simulations (LES) or Detached-Eddy Simulations (DES)
通过大涡模拟（LES）或则分离涡模拟（DES）的近场噪声的预测
Prediction of mid-field noise by acoustic wave equations, Linearized Euler Equations (LEE), or Acoustic Perturbation
Equations (APE) (see Acoustic Wave Model)
用声波方程、线性化欧拉方程组(LEE)或声学扰动方程（APE）预测中场噪声（见声波模型）
Prediction of far-field noise by the Ffowcs Williams-Hawkings equation (FWH), Boundary Element Method (BEM), Finite
Element Analysis (FEA), or Statistical Energy Analysis (SEA) (see Ffowcs Williams-Hawkings Model)
由Ffowcs Williams Hawkings方程（FWH），边界元法（BEM），有限元分
析（FEA），或统计能量分析（SEA）预测的远场噪声（见Ffowcs Williams Hawkings模型）
Contents
内容：
Broadband Noise Source Models
宽频噪声模型
Acoustic Wave Model
声波模型
Acoustic Suppression Zone Model
声抑制区模型
Ffowcs Williams-Hawkings Model
FWH模型
Aeroacoustics Bibliography
参考文献
目录
BROADBAND NOISE SOURCE MODELS 宽频噪声源模型 (9)
S PEED OF S OUND 声音的速度 (10)
C URLE N OISE S OURCE M ODEL柯尔噪声源模型 (11)
G OLDSTEIN A XISYMMETRIC N OISE S OURCE M ODEL德斯坦轴对称噪声源模型 (14)
P ROUDMAN N OISE S OURCE M ODEL 普劳德曼噪声源模型 (17)
L ILLEY N OISE S OURCE M ODEL –L ILLEY噪声源模型 (19)
L INEARIZED E ULER E QUA TION (LEE)N OISE S OURCE M ODEL 线性欧拉（LEE）噪声源模型 (22)
S TOCHASTIC N OISE G ENERA TION AND R ADIA TION (SNGR)M ODEL 随机噪声形成和辐射(SGNR)模型 (24)
L IGHTHILL S TRESS T ENSOR -L IGHTHILL应力张量 (27)
M ESH F REQUENCY C UTOFF F ORMULA TION 网格截止频率公式 (29)
ACOUSTIC WAVE MODEL 声学波模型 (31)
C ONVECTED A COUSTIC W AVE E QUA TION DUILIU 对流传热的声波方程 (33)
A COUSTIC W AVE E QUA TION WITHOUT C ONVECTION 没有对流传热的声波方程 (33)
P HYSICAL D AMPING OF A COUSTIC W AVES 声波的物理衰减 (34)
L OW-P ASS T IME F ILTERS 低通时间滤波器 (35)
L OW-P ASS S PACE F ILTER 低通空间滤波器 (37)
N ON-R EFLECTIVE B OUNDARY C ONDITION非反射边界条件 (38)
S T ANDARD N EWMARK M ETHOD 标准新标记法 (39)
N EWMARK A LPHA M ETHOD 新标记法的阿尔法方法 (39)
ACOUSTIC SUPPRESSION ZONE MODEL声抑制区模型 (41)
FFOWCS WILLIAMS-HAWKINGS MODEL -FWH模型 (45)
F FOWCS W ILLIAMS-H AWKINGS F ORMULA TION –FWH公式 (47)
FW-H S URFACE T ERMS -FWH面项 (50)
FW-H V OLUME T ERMS FW-H体积项 (55)
AEROACOUSTICS BIBLIOGRAPHY 气动声学参考文献 (57)
Broadband Noise Source Models 宽频噪声源模型
In near-field noise prediction, use broadband noise source models to compute the location and strength of the main noise sources. The noise sources consist of a volume distribution of quadrupole sources and a surface distribution of dipole sources. STAR-CCM+ offers a group of models capable of evaluating the broadband sources of noise.
在近场噪声预测中，采用宽带噪声源模型计算主要噪声源的位置和强度。

噪声源包括四极子源的体积分布和偶极子源的表面分布。

STAR-CCM+提供了一组可评价宽频噪声源的模型。

In the frequency domain, a broadband noise has a continuous spectrum, where the acoustic energy is continuously distributed at all frequencies in a given range. These models can compute the location and strength of the main sources of noise that are generated aerodynamically. They use flow field quantities from Reynolds-Averaged Navier-Stokes (RANS) Turbulence equations. Because these broadband noise models use RANS simulations, the required computational time is small.
在频域中，宽频噪声具有连续频谱，其中声能在一定频率范围内连续分布。

这些模型可以计算出空气动力学所产生的主要噪声源的位置和强度。

他们使用雷诺平均下的纳维斯托克（RANS）湍流方程的流场量。

由于这些宽带噪声模型使用RANS模拟，所需的计算时间是很小的。

In STAR-CCM+, the following broadband noise source models estimate the local acoustic power:
在STAR-CCM+中，以下的宽频噪声源模型预估局部的声功率：
For dipole-generated sound, generated per unit surface, use: 对于单位表面产生的偶极子的声音，使用：
◦ Curle noise source model
For quadrupole-generated sound, generated per unit volume, use:
对于单位体积产生的四极子的声音，使用：
◦ Goldstein axisymmetric noise source model
◦ Proudman noise source model
For sound generated by the turbulent shear flow, use:
对于湍流剪切流产生的声音，使用：
◦ Lilley noise source model
◦ Linearized Euler Equation (LEE) noise source model
To compute a synthetic turbulence field, use:
计算合成湍流场，使用：
◦ Stochastic Noise Generation and Radiation (SNGR) model
For noise sources generated by flow, use:
对于流动形成的噪声源，使用：
◦Lighthill stress tensor
For suppressing spurious reflections, use:
对于抑制伪反射，使用：
◦Acoustic Suppression Zone model
To check if the grid is fine enough to resolve turbulent flow structures of interest, use:
对检查网格是否足够精细，以解决关注的湍流结构部分，使用:
◦Mesh frequency cutoff analysis
The decomposition of the main sources of aerodynamically generated sound can be done onto a shear noise term, representing the interaction between the mean shear velocities and turbulent velocity components of the flow, and the self-noise term, representing turbulence-turbulence interactions.
对空气动力学产生的主要声源的可以分解为剪切噪声项，表示平均的剪切速度和流动的湍流速度分量之间的相互作用，以及自身噪声项，表示湍流-湍流之间的相互作用。

Speed of Sound 声音的速度
The far-field speed of sound is the distance that a sound wave travels in a unit of time.
远场声音的速度是一个声波在一个单位时间内传播的距离。

(3508)
where:
而：
is the adiabatic index or the ratio of heat capacity at constant pressure to the heat capacity at constant volume for the fluid.
是绝热指数，流体在恒压热容与定容热容的比值。

is the universal gas constant.
为通用气体常数。

is the absolute temperature, measured in Kelvin.
为绝对温度，以开始温度计量。

In dry air at 293 K, the speed of sound is 343.2 m/s.
在293 K的干空气中，生硬的速度为343.2 m/s.
Curle Noise Source Model柯尔噪声源模型
The Curle Noise source model evaluates the noise from a turbulent boundary layer flow over a solid body at low Mach number.
柯尔噪声源模型评估低马赫数的固体表面湍流边界层流动的噪声。

The model represents dipole sources of noise, that is, noise coming from the fluctuating surface pressure from solid boundaries acting on the fluid. Specifically, this model computes the local contribution per unit surface area of the body surface to the total acoustic power. This model assumes isotropic turbulence.
该模型表示噪声的偶极子源，即来自流体作用的固体边界表面压力的波动产生的噪声。

具体来说，该模型计算物体表面单位面积对总声功率的局部贡献。

该模型假定各向同性湍流。

The Curle model has been used for air handling subsystem (AHS) applications, including rotating parts (such as blowers), heat exchangers, mufflers, distribution ducts, and wall-enclosed spaces. 柯尔模型已用于空气调节系统（AHS）的应用，包括旋转部件（如鼓风机）、换热器、消声器、管道、密闭空间。

The Curle surface integral is:
柯尔曲面积分为：
(3509)
where:
而：
is the acoustic pressure.
是声压级。

is the far-field sound speed.
为远场声速。

is the emission time.
为耗散时间。

is the surface pressure.
为表面压力
is the location of the solid body.
为固体的位置。

is the point where the noise is measured.
为噪声测量点。

is .
为.
is the wall-normal direction.
为壁面法向向量。

On the assumption of small perturbations and an adiabatic problem, then:
在小扰动和绝热问题的假设下：
(3510)
where , the ratio of specific heats at constant pressure () and constant volume ().
而，恒压比热（）和定容比热（）的比值。

This value can be used to relate variations in acoustic pressure with density perturbations:
这个值可以用来将声压变化与密度扰动联系起来。

(3511)
Then Eqn. (3509) becomes:
这时方程（3509）变为：
(3512) The acoustical directional intensity along , per unit surface of the solid body on the far-field prediction, is approximated with:
声学定向强度沿着，远场预测的固体每单位表面上，近似于：
(3513
)
where:
而：
is the correlation area.
为相关区域。

is the angle between and the wall-normal direction .
为和壁面法向方向的夹角。

The measure of the local contribution to acoustic power per unit surface area, Surface Acoustic Power or SAP, can be computed from:
测量单位面积对声功率的贡献，表面声功率或（SAP）：可以从以下方程计算：
(35
14) where is the directional acoustic intensity per unit surface.
而为单位表面定向声波强度。

The model can be activated for steady and unsteady cases with Reynolds-Averaged Navier-Stokes (RANS) Turbulence models, which can provide turbulence time scale, turbulence length scale, and wall shear stress necessary to compute , the mean-square time derivative of the source surface pressure [785].
The acoustic power per unit surface can be reported in dimensional
units () and in dB:
这个模型可以从稳态和非稳态的案例中使用Reynolds-Averaged Navier-Stokes (RANS) Turbulence模型激活，这个模型可以提供必要的湍流时间
范围，湍流长度范围，和壁面剪应力来计算，源面压力均方根的
时间导数。

单位表面的声功率可以用量纲单位（）和分贝来表示：
(3515)
where is the reference acoustic power.
而为参考声功率。

Goldstein Axisymmetric Noise Source Model德斯坦轴对称噪声源模型The Goldstein model is based on Lighthill's theory for predicting aerodynamic noise from a turbulent shear flow on an assumption of locally homogeneous and anisotropic turbulence.
德斯坦模型是基于Lighthill理论，以局部均质和各向异性湍流为假设，来预测
湍流剪切流动的气动噪声。

Goldstein [784] generalized the model that Ribner [798] developed for axisymmetric turbulent flow. It eliminates the Ribner assumption of the space-time correlations of Reynolds stresses.
德斯坦[ 784 ]扩展了里布纳[ 798 ]开发的轴对称湍流模型。

它消除了雷诺应力的时空相关性的里布纳假设。

This model computes the local noise contribution per unit volume for anisotropic turbulence in axisymmetric turbulent jets.
该模型计算轴对称湍流射流中单位体积在各向异性湍流中的噪声局部贡献。

The local contribution to the total acoustic power per unit cell volume is computed from:
计算每个单元体积对总声功率的局部贡献:
(351
6) where and are the radial and angular coordinates of the receiver
location, with the receiver assumed to be sufficiently far away from the flow. is the directional acoustic intensity per unit volume of a jet:
而和接收器位置的径向和角坐标，假设接收器足够远离流动。

为射流的单位体积定向声波强度：
(3517
) is the modified convection factor:
C是修改后的对流系数：
(3518)
The decomposition of the sound intensity into shear and self-noise (with no coupling terms) occurs under the assumption that turbulence is locally homogeneous:
在湍流是局部均匀的前提下，将声强分解为剪切和自身噪声（没有耦合项）：
(3519)
(3520)
(3521)
where is the far-field density and is the far-field speed of sound.
而远场密度，为远场声速。

(35
22)
(352
3) The effect of the anisotropic structure of the turbulence appears through the following parameters:
湍流各向异性结构的影响通过以下参数呈现：
(3524)
where and are the longitudinal and transverse correlation lengths, which are not equal. The longitudinal and transversal turbulence velocities and are not equal in all directions, so and . In the limiting case of isotropic turbulence the directivity expressions become identical to those of the Ribner model, so and .
而和是纵向和横向的不相等的关联长度，纵向和横向的湍流速度和
在所有方向都不相等，这样且。

在各向同性的极限的情况下，方向性的表达式变为和里布纳模型一样，这样且。

The longitudinal and transverse turbulence velocities and are computed using the following relations:
纵向和横向的湍流速度和使用以下关系式计算：
(3525)
where:
而：
is the turbulence time scale, .
为湍流时间范围，。

is the turbulence length scale, .
为湍流长度范围，。

is the turbulent kinetic energy and is the dissipation rate.
为湍流动能，为耗散率。

In STAR-CCM+ the total acoustic power per unit volume can be
reported in dimensional units () and in dB:
在STAR-CCM+中单位体积的总声压功率可以用量纲单位（）和分贝来表示：
(3526)
where is the reference acoustic power.
而为参考声功率。

See also Bechara and others [769].
请参阅Bechara等[769]。

Proudman Noise Source Model 普劳德曼噪声源模型
Like the Goldstein model, the Proudman noise source model evaluates acoustic power per unit volume, and the sound is from quadrupoles, such as the areas around blower blades. This model has been used for air handling subsystem (AHS) applications, including rotating parts (such as blowers), heat exchangers and distribution ducts. The Proudman model assumes isotropic turbulence.
像德斯坦模型，普劳德曼噪声源模型评价从四极子发出的声音的单位体积声功率，如在风机叶片领域。

该模型已用于空气调节系统（AHS）的应用，包括旋转部件（如送风机）、热交换器和管道。

普劳德曼模型假定各向同性湍流。

Proudman considered the generation of noise by isotropic turbulence and using statistical models of various two-point moments, using the Lighthill analogy. In Proudman’s high-Reynolds model for isotropic turbulence in near incompressible flow, the AP (acoustic power) per unit volume is:
普劳德曼采用Lighthill的类推，使用不同时刻的两点统计模型，考虑各向同性湍流产生的噪音。

对于近场各向同性的不可压缩湍流中的普劳德曼高雷诺数模型，单位体积的AP（声功率）为：
(3527)
where:
而
is a constant related to the shape of the longitudinal
velocity correlation.
为与纵向速度相关性的形状有关的常数。

is the root mean square of one of the velocity components. 为一个速度分量的均方根。

is the longitudinal integral length scale of the velocity.
为速度的纵向积分的长度范围。

is the far-field density.
为远场密度。

is the far-field sound speed.
为远场的声音速度。

Different researchers have found different values for , ranging from 0.629 to 13. STAR-CCM+ uses a re-scaled constant from
Sarkar and Hussaini [800].
不同的研究者发现的不同值，范围从0.629 to 13。

STAR-CCM+使用从Sarkar 和Hussaini重新调整的。

In Proudman’s paper [797], the characteristic velocity of energy-containing eddies can be written as:
在普劳德曼论文中[797]，含涡能量的特征速度可以被写为：
(3528)
In terms of STAR-CCM+ of the turbulence velocity scale and of the turbulence length scale, the local acoustic power due to the unit volume of isotropic turbulence (in ) becomes:
依据STAR-CCM+湍流速度范围和湍流长度范围，各向同性的单位体积的局部声功率（）变为：
(3529)
with:
与：
(3530)
where:
而：
is the far-field density.
为远场密度。

is the turbulence velocity.
为湍流速度。

is the turbulence length scal.
为湍流长度范围。

is the turbulence time scale.
为湍流时间范围。

is the far-field sound speed.
C为远场声速。

The rescaled constant is based on Direct Numerical Simulation for isotropic turbulence done by Sarkar and Hussaini [800].
重新调整的常数为Sarkar 和Hussaini[800]基于各向同性湍流的直接数
值模拟。

The total acoustic power per unit volume can be reported in dimensional units () and in dB:
单位体积总声功率可以量纲单位（）和分贝来表示：
(3531)
where is the reference acoustic power.
而为参考声功率。

Lilley Noise Source Model –Lilley噪声源模型
The Lilley noise source models compute the turbulent shear flow noise of quadrupole sources from the fluctuating velocity flow field. Lilley噪声源模型从波动的流场速度计算四极子源的剪切湍流切噪声。

Applications include problems where local shear effects dominate, such as flow around solid bodies and shear flows in channels. This model evaluates the location and strength of aeroacoustic sound sources. This model is available only for steady-state simulations. 应用包括局部剪切效应作为主导的问题，例如固体绕流与通道内剪切流。

这个模型预估气动噪声源的位置和强度。

这个模型仅适用于稳态模拟。

The Lilley Noise Source model is compatible with the coupled and segregated flow models. Computational costs (specifically CPU time) depend on input of Number of Realizations and Number of Fourier Modes provided to the Stochastic Noise Generation and Radiation (SNGR) Model. The Lilley Noise Source model computes three terms: Lilley Self-Noise Source, Lilley Shear-Noise Source, and Lilley Total-Noise Source.
Lilley噪声源模型与耦合流和分离流的模型相一致。

计算时间（尤其是CPU时间）取决于个实现数和傅里叶模式数向提供的随机噪声产生和辐射（SNGR）模型的输入。

The Lilley source terms are available for steady-state solutions computed using a Reynolds-Averaged Navier-Stokes (RANS) Turbulence model. A RANS model is the only type that can provide turbulence length scale and turbulent time scale that are dominated by local shear effects, as is the case of flow around solid bodies and shear flows in channels. With this model, STAR-CCM+ provides the capability to evaluate the location and strength of aeroacoustic sound sources.
Lilley源项适用于使用Reynolds-Averaged Navier-Stokes (RANS) Turbulence进行稳态求解计算。

RANS模型是唯一一个可以提供由局部剪切效应占主导地位的湍流长度范围和湍流时间范围的类型，如同圆柱绕流和通道内剪切流动的例子一样。

与这个模型一起，STAR-CCM+提供了预测气动噪声源位置和强度的功能。

The Lilley equation can be written as:
Lilley方程可以写为：
(3532)
(3533) where is the speed of sound, is the stress tensor and
represents the effects of entropy fluctuations and fluid viscosity.
而为声速，为应力张量，表示熵波动和流体粘性效应。

The entropy fluctuation is negligible for problems without heat transfer or combustion. Viscous effects are also negligible for a high Reynolds number flow. It is therefore usual to neglect these terms from the right-hand side. Then Eqn. (3532) becomes:
如果没有传热和燃烧，熵波动是可以忽略的。

对于高雷诺数的流动，粘性效应同样可以忽略的。

因此通常忽略右边的项。

这时方程Eqn. (3532)变为：
(3534)
where:
而：
(3535)
and is reference pressure.
为参考压力。

By assuming that the flow satisfies the small perturbation assumption, the flow is locally linearized as follows:
通过假设流体满足小的扰动的假定，流动为局部的线性，如下：
(3536)
where is the mean velocity field and is the turbulent velocity field. The right-hand side of Eqn. (3534) can be expanded as follows:
而为平均速度场，为湍流速度场。

Eqn. (3534)右边的可以展开为：
(353
7)
(3538
)
(3539
)
(3540
) where represents the mean flow shape change, the Self-Noise Term contains effects of the turbulent (fluctuating) velocity components, and the Shear Noise Term contains the interaction between the mean shear flow and turbulent velocity components.
而代表平均流的形变，自身噪声项包含湍流（波动）速度部分的效应，剪切噪声项包含平均剪切流和湍流部分的相互作用。

The noise source terms of Lilley’s equation are evaluated based on mean velocity fields that are provided by a steady-state RANS solution and turbulent velocity components. These components are generated by the synthetic turbulence method of Bailly and Juve [768], Bechara et al.[769], and Kraichnan [789].
Lilley方程的噪声源项是基于稳态RANS求解和湍流速度部分提供的平均速度场来预测的。

这些部分是由Bailly和Juve [ 768 ] ,Bechara等人[ 769 ]、[ 789 ]和Kraichnan综合形成的湍流方法。

Linearized Euler Equation (LEE) Noise Source Model 线性欧拉（LEE）噪声源模型
The Linearized Euler Equation (LEE) account for refraction and convection effects in any sheared mean flows. The equations can be used for a wider range of conditions than the Lilley equation, where the associated source term is a nonlinear function of the fluctuating velocity flow field. See [769].
线性欧拉（LEE）噪声源模型可以对任何剪切平均流中的折射和对流效应做出解释。

这个方程比Lilley方程的使用条件更加宽广，它的相关的源项是一个波动速度流场的非线性函数。

This model is used for noise sources only, not for solving acoustic wave propagation.
模型仅适用于噪声源，不能解决声波的传播。

This model is available only for steady-state simulations. As Lilley equation, the LEE Noise Source model computes LEE Self-Noise Source, LEE Shear Noise Source, and LEE Total Noise Source.
这个模型仅适用于稳态模拟。

如Lilley方程，LEE噪声源模型计算LEE自身噪声源，LEE剪切噪声源和LEE总噪声源。

The linearized form of the Euler equations [769] derives from the non-linear Euler equations. The linearized forms can be achieved by superimposing small perturbations of turbulent and acoustic components on the mean flow as follows:
线性化的欧拉方程源自非线性的欧拉方程。

线性化的格式可以通过叠加平均流动的湍流和声学部分的小扰动来取得，如下：
(3541)
where the acoustic components , , are small compared to the mean components , , and the turbulent components , , .
而声学部分，，相比，，和湍流部分，，是比较小的。

Including the decomposition of variables in Eqn. (3541), and considering the assumption that the acoustic perturbations are isentropic:
包括Eqn. (3541)变量的分解，和考虑声学扰动为各向同性的假设：
(3542)
and that turbulent density fluctuations () can be neglected, then
the Linearized Euler equations for velocity components become:
和湍流密度的扰动可以忽略，这时速度部分的线性欧拉方程变为：
(3543) The second set of terms includes the source terms responsible for
sound generation.
第二个设置项包括声音形成的源项。

The decomposition of the main sources of aerodynamically
generated sound can be done onto the shear noise term
(representing the interaction between the mean shear velocities and turbulent velocity components of the flow) and the self-noise term (representing turbulence-turbulence interactions):
空气动力学形成的声音的主源的可以分解为剪切噪声项（表示流动的平均剪切速度和湍流速度部分的相互关系）和自身噪声项
（表示湍流-湍流的相互关系）:
(3544)
(3545) The turbulent velocity components are synthesized by the
Stochastic Noise Generation and Radiation model.
湍流速度部分由Stochastic Noise Generation and Radiation模型合成。

The source terms in the Linearized Euler Equations, similar to source terms in the Lilley Equation, are available for steady-state solutions computed using a Reynolds-Averaged Navier-Stokes (RANS) Turbulence model that can provide turbulence length scale and turbulence time scale.
线性欧拉方程中的源项，类似于Lilley方程中的源项，仅适用于可提供长度范围和湍流时间范围的Reynolds-Averaged Navier-Stokes (RANS) Turbulence模型的稳态求解计算。

Stochastic Noise Generation and Radiation (SNGR) Model 随机噪声形成和辐射(SGNR)模型
The scope of the SNGR model is to compute a synthetic turbulence field.
SNGR模型的范围是计算一个合成的湍流场。

A time-space turbulent velocity field is generated using random Fourier modes. This method, called the Stochastic Noise Generation and Radiation (SNGR) Model, was proposed by Kraichnan [789] and Karweit et al.[787], and afterward developed by Bailly and Juve [768], Bechara et al.[769]. In STAR-CCM+, the SNGR method,
outlined below, requires user input of two properties of the Lilley noise source model: Number of Realizations and Number of Fourier Modes. See Lilley and LEE Noise Source Properties.
使用随机傅里叶模式可以形成一个时空湍流速度场。

这种方法，被称作Stochastic Noise Generation and Radiation (SNGR)模型，被Kraichnan [789] and Karweit等人提出。

[787]，并且后来Bailly 和Juve [768], Bechara 等人[769]进行完善。

在STAR-CCM+中，SNGR方法，如下概述，要求使用者输入Lilley噪声模型的两个参数:现实数和傅里叶模式数。

见Lilley and LEE Noise Source Properties.
Consider a three-dimensional Fourier decomposition of a turbulent homogeneous isotropic field at a given point at location :
在一个给定点位置考虑一个各向同性均质湍流场的三维傅里叶分解：
(3546)
where is the t, is the wave vector, and is the complex number. 而为湍流速度场，为声波矢量，为复数。

This Fourier integral can be transformed into a limited sum of Fourier modes:
傅里叶积分可以转变为有限的傅里叶模式的和：
(3547)
where , and are the amplitude, phase and directional unit vector respectively of the -th Fourier mode associated with the wavenumber vector .
而，和分别为与波数矢量有关的第的傅里叶模式的振幅，相位和定向单位矢量。

The amplitude of each mode is computed so that the turbulence
kinetic energy corresponds to the von Karman-Pao energy spectrum for isotropic, homogeneous turbulence:
每种模式的振幅被计算，以至于各向同性均质湍流的湍动能符合von Karman-Pao能量谱:。