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第294行： − 泰勒微尺度: 介于最大和最小尺度之间的中间尺度，使惯性发生变化。泰勒微尺度不是耗散尺度，而是将能量从最大到最小无耗散地传递下去。有些文献并不认为 Taylor 微尺度是特征长度尺度，认为能量级联只包含最大和最小尺度，而后者同时包含惯性子层和粘性子层。然而，由于泰勒微尺度在波数空间的能量和动量传递中起着主导作用，因此常常用泰勒微尺度来更方便地描述”湍流”一词。+ 第304行：
第304行： − 虽然可以求出控制流体运动的 Navier-Stokes 方程的某些特殊解，但这些解在大雷诺数下对有限扰动都是不稳定的。对初始和边界条件的敏感依赖使得流体流动在时间和空间上都不规则，因此需要统计描述。俄罗斯数学家安德雷·柯尔莫哥洛夫提出了第一个湍流统计理论，基于前面提到的能量级联的概念和自相似的概念。因此，科莫微尺度以他的名字命名。现在已经知道，自相似性被打破，因此统计描述目前被修改。+ 第314行：
第314行： − 湍流的完整描述是物理学中尚未解决的问题之一。根据一个杜撰的故事，维尔纳·海森堡被问到，如果有机会，他会问上帝什么。他的回答是: “当我遇见上帝时，我要问他两个问题: 为什么是相对论？为什么是湍流？我真的相信他会为第一个问题找到答案。”类似的俏皮话也出现在 Horace Lamb 对英国科学协会的演讲中: 我现在是一个老人了，当我死去上天堂的时候，有两件事情我希望得到启迪。一个是量子电动力学，另一个是流体的湍流运动。对于前者，我相当乐观。”+ 第328行：
第328行： − 蜡烛火焰中的烟羽从层流变为湍流。雷诺数可以用来预测这种转变将在哪里发生+ 第338行：
第338行： − 在某种程度上，湍流的发生可以通过雷诺数来预测，雷诺数是流体内部的惯性力和粘性力的比值，由于流体速度的不同而受到相对内部运动的影响，在管道内部等边界面的情况下被称为边界层。类似的效果是通过引入更高速度的流体而产生的，例如来自空气中火焰的热气体。这种相对运动产生流体摩擦，这是形成湍流的一个因素。抵消这种影响的是流体的粘度，随着粘度的增加，随着更多的动能被粘性流体吸收，粘度会逐渐抑制湍流。雷诺数量化了这两种力在给定的流动条件下的相对重要性，并且是在特定情况下湍流何时发生的指南。+ 第348行：
第348行： − 这种预测湍流起始的能力是管道系统或飞机机翼等设备的重要设计工具，但雷诺数也用于流体动力学问题的缩放，并用于确定两种不同流体流动情况之间的动态相似性，例如模型飞机与其全尺寸飞机之间的动态相似性。这种标度并不总是线性的，在这两种情况下应用雷诺数可以得到标度因子。+ 第354行：
第354行： − 由于流体分子粘度的作用，动能被大量吸收的流动情况会产生层流状态。在这方面，无量纲量的雷诺数()可以作为参考。+ 第372行：
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第437行： − 虽然没有直接将无量纲雷诺数与紊流联系起来的定理，但是大于5000的雷诺数通常(但不一定)是紊流，而低雷诺数的流动通常是层流。例如，在 Poiseuille 流动中，如果雷诺数大于2040左右的临界值，湍流首先可以持续，此外，湍流通常以层流穿插，直到雷诺数大于4000时为止。+ 第440行：
第447行： − 如果物体的尺寸逐渐增大，或者流体的粘度减小，或者流体的密度增大，就会发生这种转变。+ 第464行：
第471行： − 假设对于一个二维的湍流流动，人们能够在流体中找到一个特定的点，并测量在任何给定时间通过该点的每个粒子的实际流速(v sub x / sub，v sub y / sub)}。然后人们会发现实际流速在一个平均值附近波动:+ 第483行：
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第510行： − 数学 begin { align }+ + + + + + + 第515行：
第528行： − { tau & underbrace {- rho overline { v’ y v’ x } text { experimental value } mu text { turb } frac { partial overline { x } ，;+ 第521行：
第534行： − End { align } / math+ − − 第531行：
第542行： − 定压下的热容，流体的密度，湍流粘滞系数和湍流热导率。+ 第549行：
第560行： − 理查森的湍流概念是，湍流是由不同大小的“涡流”组成的。这些大小决定了涡流的特征长度尺度，也决定了拥有属性的流速尺度和时间尺度(周转时间)。大涡是不稳定的，最终会破坏起源于大涡的小涡，初始大涡的动能被分解为起源于大涡的小涡。这些较小的涡旋也经历了同样的过程，产生了更小的涡旋，这些涡旋继承了它们的前身涡旋的能量，以此类推。通过这种方式，能量从大尺度的运动传递到小尺度，直到达到足够小的长度尺度，这样流体的粘度可以有效地将动能转化为内能。+ 第559行：
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第590行： − Math eta left ( frac ^ 3} varepsilon } right) ^ {1 / 4} ，. / math+
无编辑摘要
Rotationality :Turbulent flows have non-zero vorticity and are characterized by a strong three-dimensional vortex generation mechanism known as vortex stretching. In fluid dynamics, they are essentially vortices subjected to stretching associated with a corresponding increase of the component of vorticity in the stretching direction—due to the conservation of angular momentum. On the other hand, vortex stretching is the core mechanism on which the turbulence energy cascade relies to establish and maintain identifiable structure function. In general, the stretching mechanism implies thinning of the vortices in the direction perpendicular to the stretching direction due to volume conservation of fluid elements. As a result, the radial length scale of the vortices decreases and the larger flow structures break down into smaller structures. The process continues until the small scale structures are small enough that their kinetic energy can be transformed by the fluid's molecular viscosity into heat. Turbulent flow is always rotational and three dimensional. For example, atmospheric cyclones are rotational but their substantially two-dimensional shapes do not allow vortex generation and so are not turbulent. On the other hand, oceanic flows are dispersive but essentially non rotational and therefore are not turbulent.
Rotationality :Turbulent flows have non-zero vorticity and are characterized by a strong three-dimensional vortex generation mechanism known as vortex stretching. In fluid dynamics, they are essentially vortices subjected to stretching associated with a corresponding increase of the component of vorticity in the stretching direction—due to the conservation of angular momentum. On the other hand, vortex stretching is the core mechanism on which the turbulence energy cascade relies to establish and maintain identifiable structure function. In general, the stretching mechanism implies thinning of the vortices in the direction perpendicular to the stretching direction due to volume conservation of fluid elements. As a result, the radial length scale of the vortices decreases and the larger flow structures break down into smaller structures. The process continues until the small scale structures are small enough that their kinetic energy can be transformed by the fluid's molecular viscosity into heat. Turbulent flow is always rotational and three dimensional. For example, atmospheric cyclones are rotational but their substantially two-dimensional shapes do not allow vortex generation and so are not turbulent. On the other hand, oceanic flows are dispersive but essentially non rotational and therefore are not turbulent.
旋转性: 湍流具有非零涡度，并具有强烈的三维涡旋生成机制，即漩涡拉伸。在流体动力学中，湍流本质上是受拉伸作用的涡旋，由于角动量守恒定律，所以湍流与拉伸方向相应的涡量分量增加有关。另一方面，漩涡拉伸是湍流能量级联建立和维持可识别结构功能的核心机制。一般来说，拉伸机制意味着由于流体元的体积守恒，涡旋在垂直于拉伸方向的方向上变薄。结果表明，涡的径向长度尺度减小，较大的流动结构分解为较小的结构。这个过程一直持续到小尺度结构足够小到它们的动能可以被流体的分子粘度转化为热能。湍流通常是旋转的、三维的。例如，大气旋风是旋转的，但是它们的基本二维形状不允许涡旋的产生，因此不会产生湍流。另一方面，海流是分散的，但本质上不是旋转的，因此不是湍流。
旋转性: 湍流具有非零涡度，并具有强烈的三维涡旋生成机制，即漩涡拉伸。在流体动力学中，湍流本质上是受拉伸作用的涡旋，由于角动量守恒定律，所以湍流与拉伸方向相应的涡量分量增加有关。另一方面，漩涡拉伸是湍流能量级联建立和维持可识别结构功能的核心机制。一般来说，拉伸机制意味着由于流体元的体积守恒，涡旋在垂直于拉伸方向的方向上变薄。结果表明，涡的径向长度尺度减小，较大的流动结构分解为较小的结构。这个过程一直持续到小尺度结构足够小到它们的动能可以被流体的分子粘度转化为热能。湍流通常是旋转的、三维的。例如，大气旋风是旋转的，但是它们基本的二维形状不允许涡旋的产生，因此不会产生湍流。另一方面，海流是分散的，但本质上不是旋转的，因此不是湍流。
Kolmogorov length scales : Smallest scales in the spectrum that form the viscous sub-layer range. In this range, the energy input from nonlinear interactions and the energy drain from viscous dissipation are in exact balance. The small scales have high frequency, causing turbulence to be locally isotropic and homogeneous.
Kolmogorov length scales : Smallest scales in the spectrum that form the viscous sub-layer range. In this range, the energy input from nonlinear interactions and the energy drain from viscous dissipation are in exact balance. The small scales have high frequency, causing turbulence to be locally isotropic and homogeneous.
科尔莫戈罗夫长度尺度: 构成粘性子层范围的光谱中的最小尺度。在此范围内，非线性相互作用输入的能量与粘滞耗散输入的能量处于精确平衡状态。小尺度湍流具有高频特性，使得湍流在局部各向同性和均匀性。
'''<font color="#ff8000"> 科尔莫戈罗夫长度尺度 Kolmogorov length scales</font>''': 科尔莫戈罗夫长度尺度是构成粘性子层范围的光谱中的最小尺度。在此范围内，非线性相互作用输入的能量与粘滞耗散输入的能量处于精确平衡状态。小尺度湍流具有高频特性，使得湍流在局部具备各向同性和均匀性。
; [[Taylor microscale]]s : The intermediate scales between the largest and the smallest scales which make the inertial subrange. Taylor microscales are not dissipative scale but pass down the energy from the largest to the smallest without dissipation. Some literatures do not consider Taylor microscales as a characteristic length scale and consider the energy cascade to contain only the largest and smallest scales; while the latter accommodate both the inertial subrange and the viscous sublayer. Nevertheless, Taylor microscales are often used in describing the term "turbulence" more conveniently as these Taylor microscales play a dominant role in energy and momentum transfer in the wavenumber space.
; [[Taylor microscale]]s : The intermediate scales between the largest and the smallest scales which make the inertial subrange. Taylor microscales are not dissipative scale but pass down the energy from the largest to the smallest without dissipation. Some literatures do not consider Taylor microscales as a characteristic length scale and consider the energy cascade to contain only the largest and smallest scales; while the latter accommodate both the inertial subrange and the viscous sublayer. Nevertheless, Taylor microscales are often used in describing the term "turbulence" more conveniently as these Taylor microscales play a dominant role in energy and momentum transfer in the wavenumber space.
Taylor microscales : The intermediate scales between the largest and the smallest scales which make the inertial subrange. Taylor microscales are not dissipative scale but pass down the energy from the largest to the smallest without dissipation. Some literatures do not consider Taylor microscales as a characteristic length scale and consider the energy cascade to contain only the largest and smallest scales; while the latter accommodate both the inertial subrange and the viscous sublayer. Nevertheless, Taylor microscales are often used in describing the term "turbulence" more conveniently as these Taylor microscales play a dominant role in energy and momentum transfer in the wavenumber space.
Taylor microscales : The intermediate scales between the largest and the smallest scales which make the inertial subrange. Taylor microscales are not dissipative scale but pass down the energy from the largest to the smallest without dissipation. Some literatures do not consider Taylor microscales as a characteristic length scale and consider the energy cascade to contain only the largest and smallest scales; while the latter accommodate both the inertial subrange and the viscous sublayer. Nevertheless, Taylor microscales are often used in describing the term "turbulence" more conveniently as these Taylor microscales play a dominant role in energy and momentum transfer in the wavenumber space.
'''<font color="#ff8000"> 泰勒微尺度Taylor microscales</font>'''：泰勒微尺度是介于最大和最小尺度之间的中间尺度，它能使惯性发生变化。泰勒微尺度不是耗散尺度，而是将能量从最大到最小无耗散地传递下去。有些文献并不认为是泰勒微尺度特征长度尺度，认为能量级联只包含最大尺度和最小尺度，而后者同时包含惯性子层和粘性子层。然而，由于泰勒微尺度在波数空间的能量和动量传递中起着主导作用，因此常用泰勒微尺度来更方便地描述”湍流”一词。
Although it is possible to find some particular solutions of the Navier–Stokes equations governing fluid motion, all such solutions are unstable to finite perturbations at large Reynolds numbers. Sensitive dependence on the initial and boundary conditions makes fluid flow irregular both in time and in space so that a statistical description is needed. The Russian mathematician Andrey Kolmogorov proposed the first statistical theory of turbulence, based on the aforementioned notion of the energy cascade (an idea originally introduced by Richardson) and the concept of self-similarity. As a result, the Kolmogorov microscales were named after him. It is now known that the self-similarity is broken so the statistical description is presently modified.
Although it is possible to find some particular solutions of the Navier–Stokes equations governing fluid motion, all such solutions are unstable to finite perturbations at large Reynolds numbers. Sensitive dependence on the initial and boundary conditions makes fluid flow irregular both in time and in space so that a statistical description is needed. The Russian mathematician Andrey Kolmogorov proposed the first statistical theory of turbulence, based on the aforementioned notion of the energy cascade (an idea originally introduced by Richardson) and the concept of self-similarity. As a result, the Kolmogorov microscales were named after him. It is now known that the self-similarity is broken so the statistical description is presently modified.
虽然可以从控制流体运动的Navier-Stokes方程中求出某些特殊解，但这些解在大雷诺数下对有限扰动都是不稳定的。对初始和边界条件的敏感依赖使得流体流动在时间和空间上都不规则，因此需要统计描述。俄罗斯数学家安德雷·柯尔莫哥洛夫Andrey Kolmogorov基于前面提到的能量级联的概念和自相似的概念，提出了第一个湍流统计理论。因此，科莫微尺度以他的名字命名。因为自相似性的打破已被周知，统计描述目前已修改。
A complete description of turbulence is one of the unsolved problems in physics. According to an apocryphal story, Werner Heisenberg was asked what he would ask God, given the opportunity. His reply was: "When I meet God, I am going to ask him two questions: Why relativity? And why turbulence? I really believe he will have an answer for the first." A similar witticism has been attributed to Horace Lamb in a speech to the British Association for the Advancement of Science: "I am an old man now, and when I die and go to heaven there are two matters on which I hope for enlightenment. One is quantum electrodynamics, and the other is the turbulent motion of fluids. And about the former I am rather optimistic."
A complete description of turbulence is one of the unsolved problems in physics. According to an apocryphal story, Werner Heisenberg was asked what he would ask God, given the opportunity. His reply was: "When I meet God, I am going to ask him two questions: Why relativity? And why turbulence? I really believe he will have an answer for the first." A similar witticism has been attributed to Horace Lamb in a speech to the British Association for the Advancement of Science: "I am an old man now, and when I die and go to heaven there are two matters on which I hope for enlightenment. One is quantum electrodynamics, and the other is the turbulent motion of fluids. And about the former I am rather optimistic."
湍流的完整描述是物理学中尚未解决的问题之一。根据一个杜撰的故事，维尔纳·海森堡Werner Heisenberg被问到，如果有机会，他会问上帝什么。他的回答是: “当我遇见上帝时，我要问他两个问题: 为什么是相对论？为什么是湍流？我真的相信他会为第一个问题找到答案。”类似的俏皮话也出现在贺瑞斯·兰姆Horace Lamb英国科学协会的演讲中: 我现在是一个老人了，当我死去上天堂的时候，有两件事情我希望得到启迪。一个是量子电动力学，另一个是流体的湍流运动。对于前者，我相当乐观。”
The plume from this candle flame goes from laminar to turbulent. The Reynolds number can be used to predict where this transition will take place
The plume from this candle flame goes from laminar to turbulent. The Reynolds number can be used to predict where this transition will take place
【图4：Laminar-turbulent transition层流-湍流转变。蜡烛火焰中的烟羽从层流变为湍流。使用雷诺数可以预测这种转变将在哪里发生】
The onset of turbulence can be, to some extent, predicted by the Reynolds number, which is the ratio of inertial forces to viscous forces within a fluid which is subject to relative internal movement due to different fluid velocities, in what is known as a boundary layer in the case of a bounding surface such as the interior of a pipe. A similar effect is created by the introduction of a stream of higher velocity fluid, such as the hot gases from a flame in air. This relative movement generates fluid friction, which is a factor in developing turbulent flow. Counteracting this effect is the viscosity of the fluid, which as it increases, progressively inhibits turbulence, as more kinetic energy is absorbed by a more viscous fluid. The Reynolds number quantifies the relative importance of these two types of forces for given flow conditions, and is a guide to when turbulent flow will occur in a particular situation.
The onset of turbulence can be, to some extent, predicted by the Reynolds number, which is the ratio of inertial forces to viscous forces within a fluid which is subject to relative internal movement due to different fluid velocities, in what is known as a boundary layer in the case of a bounding surface such as the interior of a pipe. A similar effect is created by the introduction of a stream of higher velocity fluid, such as the hot gases from a flame in air. This relative movement generates fluid friction, which is a factor in developing turbulent flow. Counteracting this effect is the viscosity of the fluid, which as it increases, progressively inhibits turbulence, as more kinetic energy is absorbed by a more viscous fluid. The Reynolds number quantifies the relative importance of these two types of forces for given flow conditions, and is a guide to when turbulent flow will occur in a particular situation.
在某种程度上，湍流的发生可以通过雷诺数来预测，雷诺数是流体内部的惯性力和粘性力的比值，由于流体速度的不同而受到相对内部运动的影响，在管道内部等边界面的情况下被称为边界层。通过引入更高速度的流体能产生类似的效是，例如来自空气中火焰的热气体。这种相对运动产生流体摩擦，这是形成湍流的一个因素。抵消这种影响的是流体的粘度，随着粘度的增加，随着更多的动能被粘性流体吸收，粘度会逐渐抑制湍流。雷诺数量化了这两种力在给定的流动条件下的相对重要性，所以能预测特定情况下湍流的发生时间。
This ability to predict the onset of turbulent flow is an important design tool for equipment such as piping systems or aircraft wings, but the Reynolds number is also used in scaling of fluid dynamics problems, and is used to determine dynamic similitude between two different cases of fluid flow, such as between a model aircraft, and its full size version. Such scaling is not always linear and the application of Reynolds numbers to both situations allows scaling factors to be developed.
This ability to predict the onset of turbulent flow is an important design tool for equipment such as piping systems or aircraft wings, but the Reynolds number is also used in scaling of fluid dynamics problems, and is used to determine dynamic similitude between two different cases of fluid flow, such as between a model aircraft, and its full size version. Such scaling is not always linear and the application of Reynolds numbers to both situations allows scaling factors to be developed.
这种预测湍流起始的能力对管道系统或飞机机翼等设备的设计工具非常重要，但雷诺数也用于流体动力学问题的缩放，并用于确定两种不同流体流动情况之间的动态相似性，例如模型飞机与其全尺寸飞机之间的动态相似性。在这两种情况下应用雷诺数可以得到缩放因子，这种缩放并不总是线性。
A flow situation in which the [[kinetic energy]] is significantly absorbed due to the action of fluid molecular [[viscosity]] gives rise to a [[laminar flow]] regime. For this the dimensionless quantity the [[Reynolds number]] ({{math|Re}}) is used as a guide.
A flow situation in which the [[kinetic energy]] is significantly absorbed due to the action of fluid molecular [[viscosity]] gives rise to a [[laminar flow]] regime. For this the dimensionless quantity the [[Reynolds number]] ({{math|Re}}) is used as a guide.
A flow situation in which the kinetic energy is significantly absorbed due to the action of fluid molecular viscosity gives rise to a laminar flow regime. For this the dimensionless quantity the Reynolds number () is used as a guide.
A flow situation in which the kinetic energy is significantly absorbed due to the action of fluid molecular viscosity gives rise to a laminar flow regime. For this the dimensionless quantity the Reynolds number () is used as a guide.
由于流体分子粘度的作用，动能被大量吸收时的流动会形成层流状态。在这方面，无量纲量的雷诺数可以作为参考。
* laminar flow occurs at low Reynolds numbers, where viscous forces are dominant, and is characterized by smooth, constant fluid motion;
* laminar flow occurs at low Reynolds numbers, where viscous forces are dominant, and is characterized by smooth, constant fluid motion;
* laminar flow occurs at low Reynolds numbers, where viscous forces are dominant, and is characterized by smooth, constant fluid motion;
* 层流在雷诺数较小时发生，此时粘滞力占主导地位，其特征是平稳、恒定的流体运动
* turbulent flow occurs at high Reynolds numbers and is dominated by inertial forces, which tend to produce chaotic [[Eddy (fluid dynamics)|eddies]], [[Vortex|vortices]] and other flow instabilities.
* turbulent flow occurs at high Reynolds numbers and is dominated by inertial forces, which tend to produce chaotic [[Eddy (fluid dynamics)|eddies]], [[Vortex|vortices]] and other flow instabilities.
* turbulent flow occurs at high Reynolds numbers and is dominated by inertial forces, which tend to produce chaotic [[Eddy (fluid dynamics)|eddies]], [[Vortex|vortices]] and other flow instabilities.
* 湍流在雷诺数较大时发生，并受惯性力支配，惯性力往往会产生混乱的紊流和其他流体不稳定性。
* {{mvar|[[Rho (letter)|ρ]]}} is the [[density]] of the fluid ([[SI units]]: kg/m<sup>3</sup>)
* {{mvar|[[Rho (letter)|ρ]]}} is the [[density]] of the fluid ([[SI units]]: kg/m<sup>3</sup>)
* {{mvar|[[Rho (letter)|ρ]]}} is the [[density]] of the fluid ([[SI units]]: kg/m<sup>3</sup>)
* Rho|ρ(kg/m)是流体的密度
* {{mvar|v}} is a characteristic velocity of the fluid with respect to the object (m/s)
* {{mvar|v}} is a characteristic velocity of the fluid with respect to the object (m/s)
* {{mvar|v}} is a characteristic velocity of the fluid with respect to the object (m/s)
* v(m/s)是流体相对于物体的特征速度
* {{mvar|L}} is a characteristic linear dimension (m)
* {{mvar|L}} is a characteristic linear dimension (m)
* {{mvar|L}} is a characteristic linear dimension (m)
* L(m)是线性尺寸特征
* {{mvar|[[Mu (letter)|μ]]}} is the [[dynamic viscosity]] of the [[fluid]] (Pa·s or N·s/m<sup>2</sup> or kg/(m·s)).
* {{mvar|[[Mu (letter)|μ]]}} is the [[dynamic viscosity]] of the [[fluid]] (Pa·s or N·s/m<sup>2</sup> or kg/(m·s)).
* {{mvar|[[Mu (letter)|μ]]}} is the [[dynamic viscosity]] of the [[fluid]] (Pa·s or N·s/m<sup>2</sup> or kg/(m·s)).
* Mu|μ(Pa·s或N·s/m或kg/(m·s))是流体动态速度
While there is no theorem directly relating the non-dimensional Reynolds number to turbulence, flows at Reynolds numbers larger than 5000 are typically (but not necessarily) turbulent, while those at low Reynolds numbers usually remain laminar. In Poiseuille flow, for example, turbulence can first be sustained if the Reynolds number is larger than a critical value of about 2040; moreover, the turbulence is generally interspersed with laminar flow until a larger Reynolds number of about 4000.
While there is no theorem directly relating the non-dimensional Reynolds number to turbulence, flows at Reynolds numbers larger than 5000 are typically (but not necessarily) turbulent, while those at low Reynolds numbers usually remain laminar. In Poiseuille flow, for example, turbulence can first be sustained if the Reynolds number is larger than a critical value of about 2040; moreover, the turbulence is generally interspersed with laminar flow until a larger Reynolds number of about 4000.
虽然没有将无量纲雷诺数与紊流直接联系起来的定理，但是大于5000的雷诺数通常(但不一定)是紊流，而小雷诺数的流动通常是层流。例如，在泊肃叶流中，如果雷诺数大于2040左右的临界值，湍流可以持续，此外，湍流中通常穿插着层流，直到雷诺数大于4000时为止。
The transition occurs if the size of the object is gradually increased, or the viscosity of the fluid is decreased, or if the density of the fluid is increased.
The transition occurs if the size of the object is gradually increased, or the viscosity of the fluid is decreased, or if the density of the fluid is increased.
如果物体的尺寸逐渐增大，或流体的粘度减小，或流体的密度增大，这种转变就会发生。
Assume for a two-dimensional turbulent flow that one was able to locate a specific point in the fluid and measure the actual flow velocity (v<sub>x</sub>,v<sub>y</sub>)}} of every particle that passed through that point at any given time. Then one would find the actual flow velocity fluctuating about a mean value:
Assume for a two-dimensional turbulent flow that one was able to locate a specific point in the fluid and measure the actual flow velocity (v<sub>x</sub>,v<sub>y</sub>)}} of every particle that passed through that point at any given time. Then one would find the actual flow velocity fluctuating about a mean value:
假设人们能够在一个二维的湍流流动中找到一个特定的点，并测量在任何给定时间通过该点的每个粒子的实际流速。然后人们会发现实际流速在一个平均值上下波动:
and similarly for temperature ( + T′}}) and pressure ( + P′}}), where the primed quantities denote fluctuations superposed to the mean. This decomposition of a flow variable into a mean value and a turbulent fluctuation was originally proposed by Osborne Reynolds in 1895, and is considered to be the beginning of the systematic mathematical analysis of turbulent flow, as a sub-field of fluid dynamics. While the mean values are taken as predictable variables determined by dynamics laws, the turbulent fluctuations are regarded as stochastic variables.
and similarly for temperature ( + T′}}) and pressure ( + P′}}), where the primed quantities denote fluctuations superposed to the mean. This decomposition of a flow variable into a mean value and a turbulent fluctuation was originally proposed by Osborne Reynolds in 1895, and is considered to be the beginning of the systematic mathematical analysis of turbulent flow, as a sub-field of fluid dynamics. While the mean values are taken as predictable variables determined by dynamics laws, the turbulent fluctuations are regarded as stochastic variables.
对于温度(+ t ′})和压力(+ p ′})也类似，其中引物量表示叠加在平均值上的涨落。这种将流量变量分解为平均值和湍流波动的方法最初由奥斯鲍恩·雷诺于1895年提出，被认为是对湍流作为流体动力学的一个子领域进行系统数学分析的开始。将平均值作为由动力学规律决定的可预测变量，湍流涨落作为随机变量。
对于温度(+T)和压力(+ P′)也类似，其中引物量表示叠加在平均值上的涨落。这种将流量变量分解为平均值和湍流波动的方法最初由奥斯鲍恩·雷诺Osborne Reynolds于1895年提出，被视认将湍流作为流体动力学的一个子领域进行系统数学分析的开端。将平均值作为由动力学规律决定的可预测变量，湍流涨落作为随机变量。
<math>\begin{align}
<math>\begin{align}
开始
q&=\underbrace{v'_y \rho c_P T'}_\text{experimental value} = -k_\text{turb}\frac{\partial \overline{T}}{\partial y} \,; \\
q&=\underbrace{v'_y \rho c_P T'}_\text{experimental value} = -k_\text{turb}\frac{\partial \overline{T}}{\partial y} \,; \\
q&=\underbrace{v'_y \rho c_P T'}_\text{experimental value} = -k_\text{turb}\frac{\partial \overline{T}}{\partial y} \,; \\
q&=\underbrace{v'_y \rho c_P T'}_\text{experimental value} = -k_\text{turb}\frac{\partial \overline{T}}{\partial y} \,; \\
q&=\underbrace{v'_y \rho c_P T'}_\text{experimental value} = -k_\text{turb}\frac{\partial \overline{T}}{\partial y} \
Q & underbrace { v’ y rho c p t’} text { experimental value }-k { turb } frac { partial overline { t } ，;
Q & underbrace { v’ y rho c p t’} text { experimental value }-k { turb } frac { partial overline { t } ，;
Q & underbrace { v’ y rho c p t’} text { experimental value }-k { turb } frac { partial overline { t } ，
Q & underbrace { v’ y rho c p t’} text { experimental value }-k { turb } frac { partial overline { t } ，
\tau &=\underbrace{-\rho \overline{v'_y v'_x}}_\text{experimental value} = \mu_\text{turb}\frac{\partial \overline{v}_x}{\partial y} \,;
\tau &=\underbrace{-\rho \overline{v'_y v'_x}}_\text{experimental value} = \mu_\text{turb}\frac{\partial \overline{v}_x}{\partial y} \,;
\tau &=\underbrace{-\rho \overline{v'_y v'_x}}_\text{experimental value} = \mu_\text{turb}\frac{\partial \overline{v}_x}{\partial y} \,;
\tau &=\underbrace{-\rho \overline{v'_y v'_x}}_\text{experimental value} = \mu_\text{turb}\frac{\partial \overline{v}_x}{\partial y} \,;
\tau &=\underbrace{-\rho \overline{v'_y v'_x}}_\text{experimental value} = \mu_\text{turb}\frac{\partial \overline{v}_x}{\partial y} \,;
\end{align}</math>
\end{align}</math>
\end{align}</math>
\end{align}</math>
结束
where is the heat capacity at constant pressure, is the density of the fluid, is the coefficient of turbulent viscosity and is the turbulent thermal conductivity.
where is the heat capacity at constant pressure, is the density of the fluid, is the coefficient of turbulent viscosity and is the turbulent thermal conductivity.
其中分别是恒定压力下的热容、流体的密度、湍流粘度的系数、湍流热导率
Richardson's notion of turbulence was that a turbulent flow is composed by "eddies" of different sizes. The sizes define a characteristic length scale for the eddies, which are also characterized by flow velocity scales and time scales (turnover time) dependent on the length scale. The large eddies are unstable and eventually break up originating smaller eddies, and the kinetic energy of the initial large eddy is divided into the smaller eddies that stemmed from it. These smaller eddies undergo the same process, giving rise to even smaller eddies which inherit the energy of their predecessor eddy, and so on. In this way, the energy is passed down from the large scales of the motion to smaller scales until reaching a sufficiently small length scale such that the viscosity of the fluid can effectively dissipate the kinetic energy into internal energy.
Richardson's notion of turbulence was that a turbulent flow is composed by "eddies" of different sizes. The sizes define a characteristic length scale for the eddies, which are also characterized by flow velocity scales and time scales (turnover time) dependent on the length scale. The large eddies are unstable and eventually break up originating smaller eddies, and the kinetic energy of the initial large eddy is divided into the smaller eddies that stemmed from it. These smaller eddies undergo the same process, giving rise to even smaller eddies which inherit the energy of their predecessor eddy, and so on. In this way, the energy is passed down from the large scales of the motion to smaller scales until reaching a sufficiently small length scale such that the viscosity of the fluid can effectively dissipate the kinetic energy into internal energy.
理查森Richardson的湍流概念是，湍流是由不同大小的“涡流”组成的。这些大小决定了涡流的特征长度尺度，也决定了拥有属性的流速尺度和时间尺度(周转时间)。大涡是不稳定的，最终会破坏起源于大涡的小涡，初始大涡的动能被分解为来源于大涡的小涡。这些较小的涡旋也经历了同样的过程，产生了更小的涡旋，这些涡旋继承了它们的前身涡旋的能量，以此类推。通过这种方式，能量从大尺度的运动传递到小尺度，直到达到足够小的长度尺度，这样流体的粘度可以有效地将动能转化为内能。
In his original theory of 1941, Kolmogorov postulated that for very high Reynolds numbers, the small-scale turbulent motions are statistically isotropic (i.e. no preferential spatial direction could be discerned). In general, the large scales of a flow are not isotropic, since they are determined by the particular geometrical features of the boundaries (the size characterizing the large scales will be denoted as ). Kolmogorov's idea was that in the Richardson's energy cascade this geometrical and directional information is lost, while the scale is reduced, so that the statistics of the small scales has a universal character: they are the same for all turbulent flows when the Reynolds number is sufficiently high.
In his original theory of 1941, Kolmogorov postulated that for very high Reynolds numbers, the small-scale turbulent motions are statistically isotropic (i.e. no preferential spatial direction could be discerned). In general, the large scales of a flow are not isotropic, since they are determined by the particular geometrical features of the boundaries (the size characterizing the large scales will be denoted as ). Kolmogorov's idea was that in the Richardson's energy cascade this geometrical and directional information is lost, while the scale is reduced, so that the statistics of the small scales has a universal character: they are the same for all turbulent flows when the Reynolds number is sufficiently high.
在1941年最初的理论中，Kolmogorov 假定对于很高的雷诺数，小尺度的湍流运动在统计上是各向同性的(即:。没有优先的空间方向可以辨别)。一般来说，大尺度的流动并不是各向同性的，因为它们是由边界的特殊几何特征决定的(表征大尺度的尺寸将被表示为)。科尔莫戈罗夫的想法是，在理查森的能量级联中，这种几何和方向的信息丢失了，而尺度缩小了，因此小尺度的统计具有一个普遍的特征: 当雷诺数足够高时，它们对所有湍流都是一样的。
在1941年最初的理论中，Kolmogorov 假定对于很高的雷诺数，小尺度的湍流运动在统计上是各向同性的(即没有优先的空间方向可以辨别)。一般来说，大尺度的流动并不是各向同性的，因为它们是由边界的特殊几何特征决定的(表征大尺度的尺寸将被表示为)。科尔莫戈罗夫的想法是，在 Richardson的能量级联中，虽然尺度缩小了，但这种几何和方向的信息丢失了，，因此小尺度的统计具有一个普遍的特征: 当雷诺数足够高时，它们对所有湍流都是一样的。
<math>\eta = \left(\frac{\nu^3}{\varepsilon}\right)^{1/4} \,.</math>
<math>\eta = \left(\frac{\nu^3}{\varepsilon}\right)^{1/4} \,.</math>
\eta = \left(\frac{\nu^3}{\varepsilon}\right)^{1/4} \,