# 湍流

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In fluid dynamics, **turbulence** or **turbulent flow** is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between those layers.^{[1]}

在流体力学（fluid dynamics）中，湍流或紊流是以压力和流速（flow velocity）的混乱变化为特征的流体运动。相反，层流（ laminar flow）是指流体在无互相干扰和中断的平行层中流动。

Turbulence is commonly observed in everyday phenomena such as surf, fast flowing rivers, billowing storm clouds, or smoke from a chimney, and most fluid flows occurring in nature or created in engineering applications are turbulent.^{[2]}^{[3]}^{:2} Turbulence is caused by excessive kinetic energy in parts of a fluid flow, which overcomes the damping effect of the fluid's viscosity. For this reason turbulence is commonly realized in low viscosity fluids. In general terms, in turbulent flow, unsteady vortices appear of many sizes which interact with each other, consequently drag due to friction effects increases. This increases the energy needed to pump fluid through a pipe. Turbulence can be exploited, for example, by devices such as aerodynamic spoilers on aircraft that "spoil" the laminar flow to increase drag and reduce lift.

可以在日常现象中观察到湍流，如拍岸海浪（surf）、快速流动的的河水、翻腾的风暴云或从烟囱中冒出的烟。大多数在自然界发生或在工程应用中产生的流体运动都是湍流。湍流是由于流体的部分动能过大，克服了流体粘度的阻尼效应而引起的。因此，湍流通常在低粘度流体中出现。一般来说，湍流中会出现许多大小不一的不稳定涡流（vortices），它们相互作用，从而增加了摩擦效应产生的阻力（drag），也增加了泵送流体通过管道所需的能量。湍流可以被利用，例如，飞机上的空气动力扰流器（spoilers）等装置可以 "破坏 "层流以增加阻力和减少升力。

The onset of turbulence can be predicted by the dimensionless Reynolds number, the ratio of kinetic energy to viscous damping in a fluid flow. However, turbulence has long resisted detailed physical analysis, and the interactions within turbulence create a very complex phenomenon. Richard Feynman has described turbulence as the most important unsolved problem in classical physics.^{[4]}

湍流的发生可以通过无量纲雷诺数（Reynolds number）来预测，雷诺数是流体动能与粘性阻尼的比率。然而，对湍流的详细物理学分析难以开展，且湍流内部的相互作用非常复杂。理查德-费曼（Richard Feynman ）将湍流形容为经典物理学中最重要的未解决之问题。

## Examples of turbulence

湍流的例子

【图1：Laminar and turbulent water flow over the hull of a submarine. As the relative velocity of the water increases turbulence occurs. 潜艇船体外，水的层流和紊流。湍流随着水的相对速度增加而出现。】

【图2：tip vortex from an airplane wing飞机机翼上的翼尖涡流】

- Smoke rising from a cigarette. For the first few centimeters, the smoke is laminar. The smoke plume becomes turbulent as its Reynolds number increases with increases in flow velocity and characteristic lengthscale.

- 烟雾从一支香烟上升起。最初的几厘米，烟雾是层流。
**雷诺数（Reynolds number ）**随着流速和特征长度增加，烟雾羽流（plume）变成了湍流。

- Flow over a golf ball. (This can be best understood by considering the golf ball to be stationary, with air flowing over it.) If the golf ball were smooth, the boundary layer flow over the front of the sphere would be laminar at typical conditions. However, the boundary layer would separate early, as the pressure gradient switched from favorable (pressure decreasing in the flow direction) to unfavorable (pressure increasing in the flow direction), creating a large region of low pressure behind the ball that creates high form drag. To prevent this, the surface is dimpled to perturb the boundary layer and promote turbulence. This results in higher skin friction, but it moves the point of boundary layer separation further along, resulting in lower drag.

- 流体流过高尔夫球。 （假设高尔夫球静止，而空气在上面流动，最容易的理解。）如果高尔夫球是光滑的，则在典型条件下，球体前部的
**边界层（ boundary layer ）**会出现层流。 但是，因为压力梯度会从顺压（压力沿流动方向减小）切换到逆压（压力沿流动方向增大），边界层会提前分离，从而在球后形成一个大低压区，产生较高的型阻（form drag）。 为了避免这种情况，可以在球的表面制造凹槽以扰动边界层并促进湍流。 这会产生较高的表面摩擦，但会进一步移动边界层分离点，从而导致阻力减少。

- Clear-air turbulence experienced during airplane flight, as well as poor astronomical seeing (the blurring of images seen through the atmosphere).

- 飞机飞行时经历过
**晴空湍流（Clear-air turbulence）**，以及**天文视宁度（astronomical seeing）**不佳（通过大气看到的图像模糊）。

- Most of the terrestrial atmospheric circulation.

- Most of the terrestrial atmospheric circulation.

- 大部分陆地的大气环流

- The oceanic and atmospheric mixed layers and intense oceanic currents.

- The oceanic and atmospheric mixed layers and intense oceanic currents.

- 海洋和大气混合层和强烈的洋流

- The flow conditions in many industrial equipment (such as pipes, ducts, precipitators, gas scrubbers, dynamic scraped surface heat exchangers, etc.) and machines (for instance, internal combustion engines and gas turbines).

- The flow conditions in many industrial equipment (such as pipes, ducts, precipitators, gas scrubbers, dynamic scraped surface heat exchangers, etc.) and machines (for instance, internal combustion engines and gas turbines).

- 许多工业设备(如管道、管道、除尘器、气体洗涤器、动态刮面热交换器等)和机器(如
**内燃机internal combustion engine**、**燃气轮机gas turbine**)中的流动状况。

- The external flow over all kinds of vehicles such as cars, airplanes, ships, and submarines.

- The external flow over all kinds of vehicles such as cars, airplanes, ships, and submarines.

- 各种交通工具，如汽车、飞机、船舶和潜艇的外部流。

- The motions of matter in stellar atmospheres.

- The motions of matter in stellar atmospheres.

- 恒星大气中物质的运动。

- A jet exhausting from a nozzle into a quiescent fluid. As the flow emerges into this external fluid, shear layers originating at the lips of the nozzle are created. These layers separate the fast moving jet from the external fluid, and at a certain critical Reynolds number they become unstable and break down to turbulence.

- A jet exhausting from a nozzle into a quiescent fluid. As the flow emerges into this external fluid, shear layers originating at the lips of the nozzle are created. These layers separate the fast moving jet from the external fluid, and at a certain critical Reynolds number they become unstable and break down to turbulence.

- 从喷嘴排放到静止流体中的射流。 当流体流入该外部流体中时，会在喷嘴的边缘产生剪切层。 这些层将快速移动的射流与外部流体分开，并且在某个临界雷诺数时，它们变得不稳定并分解为湍流。

- Biologically generated turbulence resulting from swimming animals affects ocean mixing.
^{[5]}

- Biologically generated turbulence resulting from swimming animals affects ocean mixing.

- 游泳动物引起的生物湍流会影响海洋混合。

- Snow fences work by inducing turbulence in the wind, forcing it to drop much of its snow load near the fence.

- Snow fences work by inducing turbulence in the wind, forcing it to drop much of its snow load near the fence.

- 防雪栅栏的工作原理是在风中产生湍流，迫使其将大部分雪荷载降到栅栏附近。

- Bridge supports (piers) in water. In the late summer and fall, when river flow is slow, water flows smoothly around the support legs. In the spring, when the flow is faster, a higher Reynolds number is associated with the flow. The flow may start off laminar but is quickly separated from the leg and becomes turbulent.

- Bridge supports (piers) in water. In the late summer and fall, when river flow is slow, water flows smoothly around the support legs. In the spring, when the flow is faster, a higher Reynolds number is associated with the flow. The flow may start off laminar but is quickly separated from the leg and becomes turbulent.

- 桥在水中支撑码头。 在夏末和秋季，当河流流量缓慢时，水在支柱周围顺畅流动。 在春季，水流更快时，较高的雷诺数与流动相关联。 流动可能从层流开始，但很快与支柱分离并变得湍流。

- In many geophysical flows (rivers, atmospheric boundary layer), the flow turbulence is dominated by the coherent structures and turbulent events. A turbulent event is a series of turbulent fluctuations that contain more energy than the average flow turbulence.
^{[6]}^{[7]}The turbulent events are associated with coherent flow structures such as eddies and turbulent bursting, and they play a critical role in terms of sediment scour, accretion and transport in rivers as well as contaminant mixing and dispersion in rivers and estuaries, and in the atmosphere.

- In many geophysical flows (rivers, atmospheric boundary layer), the flow turbulence is dominated by the coherent structures and turbulent events. A turbulent event is a series of turbulent fluctuations that contain more energy than the average flow turbulence.

- 在许多地球物理流动(河流、大气边界层)中，湍流主要由凝聚结构和湍流事件所控制。湍流事件是一系列包含比平均湍流更多能量的湍流波动。

- In the medical field of cardiology, a stethoscope is used to detect heart sounds and bruits, which are due to turbulent blood flow. In normal individuals, heart sounds are a product of turbulent flow as heart valves close. However, in some conditions turbulent flow can be audible due to other reasons, some of them pathological. For example, in advanced atherosclerosis, bruits (and therefore turbulent flow) can be heard in some vessels that have been narrowed by the disease process.

- In the medical field of cardiology, a stethoscope is used to detect heart sounds and bruits, which are due to turbulent blood flow. In normal individuals, heart sounds are a product of turbulent flow as heart valves close. However, in some conditions turbulent flow can be audible due to other reasons, some of them pathological. For example, in advanced atherosclerosis, bruits (and therefore turbulent flow) can be heard in some vessels that have been narrowed by the disease process.

- 在心脏病学的医学领域中，听诊器被用于检测心音和杂音，这些都是由于血液湍流引起的。 在正常个体中，心音是心脏瓣膜关闭时湍流的产物。 但是，在某些情况下，由于其他原因，某些是病理性原因，也可以听到心音或杂音。 例如，在
**晚期动脉粥样硬化atherosclerosis**中，在某些因疾病过程而变窄的血管中会产生湍流，因此会听到杂音。

- Recently, turbulence in porous media became a highly debated subject.
^{[8]}

- Recently, turbulence in porous media became a highly debated subject.

- 最近，多孔介质中的湍流成为一个备受争议的话题

## Features

特征

laser-induced fluorescence. The jet exhibits a wide range of length scales, an important characteristic of turbulent flows.]]

【图3: False color image of the far field of a submerged turbulent jet. 紊乱淹没射流远场的伪彩色图像。射流的长度范围很广，这是湍流的一个重要特征激光诱导荧光。】

Turbulence is characterized by the following features:

Turbulence is characterized by the following features:

湍流的拥有如下特征:

- Irregularity
- Turbulent flows are always highly irregular. For this reason, turbulence problems are normally treated statistically rather than deterministically. Turbulent flow is chaotic. However, not all chaotic flows are turbulent.

Irregularity : Turbulent flows are always highly irregular. For this reason, turbulence problems are normally treated statistically rather than deterministically. Turbulent flow is chaotic. However, not all chaotic flows are turbulent.

不规则性: 湍流总是高度不规则的。由于这个原因，湍流问题通常用统计的方法而不是决定性地处理。湍流是紊乱的。然而，并非所有的紊流都是如此。

- Diffusivity
- The readily available supply of energy in turbulent flows tends to accelerate the homogenization (mixing) of fluid mixtures. The characteristic which is responsible for the enhanced mixing and increased rates of mass, momentum and energy transports in a flow is called "diffusivity".
^{[9]}

Diffusivity :The readily available supply of energy in turbulent flows tends to accelerate the homogenization (mixing) of fluid mixtures. The characteristic which is responsible for the enhanced mixing and increased rates of mass, momentum and energy transports in a flow is called "diffusivity".

扩散系数: 在湍流中能轻易获得稳定的能量供应，这种能量会加速流体混合物的均匀化(混合)。在流动中，增强混合和提高质量、动量和能量输送速率的特性称为“扩散系数”。

*Turbulent diffusion* is usually described by a turbulent diffusion coefficient. This turbulent diffusion coefficient is defined in a phenomenological sense, by analogy with the molecular diffusivities, but it does not have a true physical meaning, being dependent on the flow conditions, and not a property of the fluid itself. In addition, the turbulent diffusivity concept assumes a constitutive relation between a turbulent flux and the gradient of a mean variable similar to the relation between flux and gradient that exists for molecular transport. In the best case, this assumption is only an approximation. Nevertheless, the turbulent diffusivity is the simplest approach for quantitative analysis of turbulent flows, and many models have been postulated to calculate it. For instance, in large bodies of water like oceans this coefficient can be found using Richardson's four-third power law and is governed by the random walk principle. In rivers and large ocean currents, the diffusion coefficient is given by variations of Elder's formula.

Turbulent diffusion is usually described by a turbulent diffusion coefficient. This turbulent diffusion coefficient is defined in a phenomenological sense, by analogy with the molecular diffusivities, but it does not have a true physical meaning, being dependent on the flow conditions, and not a property of the fluid itself. In addition, the turbulent diffusivity concept assumes a constitutive relation between a turbulent flux and the gradient of a mean variable similar to the relation between flux and gradient that exists for molecular transport. In the best case, this assumption is only an approximation. Nevertheless, the turbulent diffusivity is the simplest approach for quantitative analysis of turbulent flows, and many models have been postulated to calculate it. For instance, in large bodies of water like oceans this coefficient can be found using Richardson's four-third power law and is governed by the random walk principle. In rivers and large ocean currents, the diffusion coefficient is given by variations of Elder's formula.

湍流扩散通常用湍流扩散系数来描述。这种湍流扩散系数是类比于分子扩散系数，从唯象的意义上定义的，但它取决于流动条件，而不是流体本身的性质，所以没有真正的物理意义。此外，湍流扩散率概念假定了湍流通量和平均变量梯度之间的本构关系，类似于分子输运中存在的通量和梯度之间的关系。在最好的情况下，这个假设只是一个近似值。然而，湍流扩散系数是定量分析湍流流动的最简单的方法，许多模型已被假定用来计算它。例如，在像海洋这样的大型水体中，这个系数可以用理查森的四分之三次方定律找到，并受随机游动原理支配。在河流和大洋流中，扩散系数是通过埃尔德公式的变化得到的。

- 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.
^{[10]}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.^{[11]}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.^{[12]}

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。

- Dissipation
- To sustain turbulent flow, a persistent source of energy supply is required because turbulence dissipates rapidly as the kinetic energy is converted into internal energy by viscous shear stress. Turbulence causes the formation of eddies of many different length scales. Most of the kinetic energy of the turbulent motion is contained in the large-scale structures. The energy "cascades" from these large-scale structures to smaller scale structures by an inertial and essentially inviscid mechanism. This process continues, creating smaller and smaller structures which produces a hierarchy of eddies. Eventually this process creates structures that are small enough that molecular diffusion becomes important and viscous dissipation of energy finally takes place. The scale at which this happens is the Kolmogorov length scale.

Dissipation : To sustain turbulent flow, a persistent source of energy supply is required because turbulence dissipates rapidly as the kinetic energy is converted into internal energy by viscous shear stress. Turbulence causes the formation of eddies of many different length scales. Most of the kinetic energy of the turbulent motion is contained in the large-scale structures. The energy "cascades" from these large-scale structures to smaller scale structures by an inertial and essentially inviscid mechanism. This process continues, creating smaller and smaller structures which produces a hierarchy of eddies. Eventually this process creates structures that are small enough that molecular diffusion becomes important and viscous dissipation of energy finally takes place. The scale at which this happens is the Kolmogorov length scale.

耗散: 湍流的维持需要一个持久的能量供应源，因为当动能通过** 粘性切应力viscous shear stress **转化为内能时，湍流会迅速消散。湍流导致了许多不同长度尺度的涡流的形成。湍流运动的大部分动能都包含在大尺度结构中。能量“级联”从这些大规模的结构到更小的规模的结构通过一个惯性和本质上无粘机制。这个过程还在继续，形成了越来越小的结构，产生了一系列的涡流。最终，这个过程创造了足够小的结构，这使分子扩散变得重要，粘性能量耗散最终发生。发生这种情况的尺度是科尔莫哥罗夫长度尺度。

Via this energy cascade, turbulent flow can be realized as a superposition of a spectrum of flow velocity fluctuations and eddies upon a mean flow. The eddies are loosely defined as coherent patterns of flow velocity, vorticity and pressure. Turbulent flows may be viewed as made of an entire hierarchy of eddies over a wide range of length scales and the hierarchy can be described by the energy spectrum that measures the energy in flow velocity fluctuations for each length scale (wavenumber). The scales in the energy cascade are generally uncontrollable and highly non-symmetric. Nevertheless, based on these length scales these eddies can be divided into three categories.

Via this energy cascade, turbulent flow can be realized as a superposition of a spectrum of flow velocity fluctuations and eddies upon a mean flow. The eddies are loosely defined as coherent patterns of flow velocity, vorticity and pressure. Turbulent flows may be viewed as made of an entire hierarchy of eddies over a wide range of length scales and the hierarchy can be described by the energy spectrum that measures the energy in flow velocity fluctuations for each length scale (wavenumber). The scales in the energy cascade are generally uncontrollable and highly non-symmetric. Nevertheless, based on these length scales these eddies can be divided into three categories.

通过这种能量级联，湍流可以通过实一个平均流上流速起伏和涡流谱的叠加实现。这些涡流被粗略地定义为流速、涡量和压力的相干模式。湍流可以视为由一个完整的长度尺度范围内的涡流层次组成的，这个层次可以用能量谱来描述，能量谱测量每个长度尺度的流速波动中的能量(波数)。能量级联中的尺度通常是不可控的，高度也不对称。然而，根据这些长度尺度，这些涡旋可以分为三类。

- Integral time scale

Integral time scale

积分时间表

The integral time scale for a Lagrangian flow can be defined as:

The integral time scale for a Lagrangian flow can be defined as:

拉格朗日流的积分时间尺度可以定义为:

- [math]\displaystyle{ T = \left ( \frac{1}{\langle u'u'\rangle} \right )\int_0^\infty \langle u'u'(\tau)\rangle \, d\tau }[/math]

```
[math]\displaystyle{ T = \left ( \frac{1}{\langle u'u'\rangle} \right )\int_0^\infty \langle u'u'(\tau)\rangle \, d\tau }[/math]
```

T = \left ( \frac{1}{\langle u'u'\rangle} \right )\int_0^\infty \langle u'u'(\tau)\rangle \, d\tau

where *u*′ is the velocity fluctuation, and [math]\displaystyle{ \tau }[/math] is the time lag between measurements.^{[13]}

where u′ is the velocity fluctuation, and [math]\displaystyle{ \tau }[/math] is the time lag between measurements.

其中u&prime是速度波动，而tau是两次测量之间的时间差。

- Integral length scales

Integral length scales

积分长度尺度

- Large eddies obtain energy from the mean flow and also from each other. Thus, these are the energy production eddies which contain most of the energy. They have the large flow velocity fluctuation and are low in frequency. Integral scales are highly anisotropic and are defined in terms of the normalized two-point flow velocity correlations. The maximum length of these scales is constrained by the characteristic length of the apparatus. For example, the largest integral length scale of pipe flow is equal to the pipe diameter. In the case of atmospheric turbulence, this length can reach up to the order of several hundreds kilometers.: The integral length scale can be defined as

Large eddies obtain energy from the mean flow and also from each other. Thus, these are the energy production eddies which contain most of the energy. They have the large flow velocity fluctuation and are low in frequency. Integral scales are highly anisotropic and are defined in terms of the normalized two-point flow velocity correlations. The maximum length of these scales is constrained by the characteristic length of the apparatus. For example, the largest integral length scale of pipe flow is equal to the pipe diameter. In the case of atmospheric turbulence, this length can reach up to the order of several hundreds kilometers.: The integral length scale can be defined as

大涡流从平均流动中获得能量，也从彼此之间获得能量。因此，它们就是包含大部分能量的能量产生涡旋。它们流速波动大，频率低。积分尺度根据归一化两点流速相关性来定义，具有高度各向异性。这些涡流的最大长度受仪器特征长度的限制。例如，管流的最大积分长度标度等于管径。在大气湍流的情况下，这个长度可以达到几百公里。整数长度标度可定义为

- [math]\displaystyle{ L = \left ( \frac{1}{\langle u'u'\rangle} \right ) \int_0^\infty \langle u'u'(r)\rangle \, dr }[/math]

```
[math]\displaystyle{ L = \left ( \frac{1}{\langle u'u'\rangle} \right ) \int_0^\infty \langle u'u'(r)\rangle \, dr }[/math]
```

L = \left ( \frac{1}{\langle u'u'\rangle} \right ) \int_0^\infty \langle u'u'(r)\rangle \, dr

- where
*r*is the distance between two measurement locations, and*u*′ is the velocity fluctuation in that same direction.^{[13]}

where r is the distance between two measurement locations, and u′ is the velocity fluctuation in that same direction.

其中 r 是两个测量点之间的距离，u & prime 是同一方向上的速度波动。

- 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 Kolmogorov length scales**: 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 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.

** 泰勒微尺度Taylor microscales**：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.^{[14]}

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."^{[15]} 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."^{[16]}^{[17]}

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。”

## Onset of turbulence

湍流开始

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.^{[18]}

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 (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.

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。

With respect to laminar and turbulent flow regimes:

With respect to laminar and turbulent flow regimes:

关于层流和湍流流型:

- 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 eddies, vortices and other flow instabilities.

- turbulent flow occurs at high Reynolds numbers and is dominated by inertial forces, which tend to produce chaotic eddies, 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。

The Reynolds number is defined as^{[19]}

The Reynolds number is defined as

雷诺数定义为

- [math]\displaystyle{ \mathrm{Re} = \frac{\rho v L}{\mu} \,, }[/math]

[math]\displaystyle{ \mathrm{Re} = \frac{\rho v L}{\mu} \,, }[/math]

mathrm{Re} = \frac{\rho v L}{\mu}

where:

where:

其中:

- Rho|ρ(kg/m)是流体的密度

- v is a characteristic velocity of the fluid with respect to the object (m/s)

- 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

- L is a characteristic linear dimension (m)

- L is a characteristic linear dimension (m)

- L(m)是线性尺寸特征

- μ is the dynamic viscosity of the fluid (Pa·s or N·s/m
^{2}or kg/(m·s)).

- μ is the dynamic viscosity of the fluid (Pa·s or N·s/m
^{2}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;^{[20]} 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。

## Heat and momentum transfer

热量和动量的传递

When flow is turbulent, particles exhibit additional transverse motion which enhances the rate of energy and momentum exchange between them thus increasing the heat transfer and the friction coefficient.

When flow is turbulent, particles exhibit additional transverse motion which enhances the rate of energy and momentum exchange between them thus increasing the heat transfer and the friction coefficient.

当流动为湍流时，颗粒表现出附加的横向运动，从而增加了颗粒间的能量和动量交换率，从而增加了传热和摩擦系数。

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** = (*v _{x}*,

*v*) 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:

_{y}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_{x},v_{y})}} 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:

- [math]\displaystyle{ v_x = \underbrace{\overline{v}_x}_\text{mean value} + \underbrace{v'_x}_\text{fluctuation} \quad \text{and} \quad v_y=\overline{v}_y + v'_y \,; }[/math]

[math]\displaystyle{ v_x = \underbrace{\overline{v}_x}_\text{mean value} + \underbrace{v'_x}_\text{fluctuation} \quad \text{and} \quad v_y=\overline{v}_y + v'_y \,; }[/math]

v_x = \underbrace{\overline{v}_x}_\text{mean value} + \underbrace{v'_x}_\text{fluctuation} \quad \text{and} \quad v_y=\overline{v}_y + v'_y \,

and similarly for temperature (*T* = 模板:Overline + *T′*) and pressure (*P* = 模板:Overline + *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。

The heat flux and momentum transfer (represented by the shear stress τ) in the direction normal to the flow for a given time are

The heat flux and momentum transfer (represented by the shear stress ) in the direction normal to the flow for a given time are

在给定的时间内，热通量和动量在法向流动方向上的传递(用切应力表示)是

- [math]\displaystyle{ \begin{align} \lt math\gt \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 { 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} \,; \end{align} }[/math]

\end{align}</math>

结束

where c_{P} is the heat capacity at constant pressure, ρ is the density of the fluid, *μ*_{turb} is the coefficient of turbulent viscosity and *k*_{turb} is the turbulent thermal conductivity.^{[3]}

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

## Kolmogorov's theory of 1941模板:Anchor

科尔莫哥罗夫1941年的理论

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 L). 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 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。

Thus, Kolmogorov introduced a second hypothesis: for very high Reynolds numbers the statistics of small scales are universally and uniquely determined by the kinematic viscosity ν and the rate of energy dissipation ε. With only these two parameters, the unique length that can be formed by dimensional analysis is

Thus, Kolmogorov introduced a second hypothesis: for very high Reynolds numbers the statistics of small scales are universally and uniquely determined by the kinematic viscosity and the rate of energy dissipation . With only these two parameters, the unique length that can be formed by dimensional analysis is

因此，Kolmogorov 提出了第二个假设: 对于很高的雷诺数，小尺度的统计量是由运动粘度和能量耗散率普遍而唯一地决定的。只有这两个参数，量纲分析可以形成的唯一长度是

- [math]\displaystyle{ \eta = \left(\frac{\nu^3}{\varepsilon}\right)^{1/4} \,. }[/math]

[math]\displaystyle{ \eta = \left(\frac{\nu^3}{\varepsilon}\right)^{1/4} \,. }[/math]

\eta = \left(\frac{\nu^3}{\varepsilon}\right)^{1/4} \,

This is today known as the Kolmogorov length scale (see Kolmogorov microscales).

This is today known as the Kolmogorov length scale (see Kolmogorov microscales).

这就是我们今天所知道的科尔莫哥罗夫长度尺度(见科莫微尺度)。

A turbulent flow is characterized by a hierarchy of scales through which the energy cascade takes place. Dissipation of kinetic energy takes place at scales of the order of Kolmogorov length η, while the input of energy into the cascade comes from the decay of the large scales, of order L. These two scales at the extremes of the cascade can differ by several orders of magnitude at high Reynolds numbers. In between there is a range of scales (each one with its own characteristic length r) that has formed at the expense of the energy of the large ones. These scales are very large compared with the Kolmogorov length, but still very small compared with the large scale of the flow (i.e. *η* ≪ *r* ≪ *L*). Since eddies in this range are much larger than the dissipative eddies that exist at Kolmogorov scales, kinetic energy is essentially not dissipated in this range, and it is merely transferred to smaller scales until viscous effects become important as the order of the Kolmogorov scale is approached. Within this range inertial effects are still much larger than viscous effects, and it is possible to assume that viscosity does not play a role in their internal dynamics (for this reason this range is called "inertial range").

A turbulent flow is characterized by a hierarchy of scales through which the energy cascade takes place. Dissipation of kinetic energy takes place at scales of the order of Kolmogorov length , while the input of energy into the cascade comes from the decay of the large scales, of order . These two scales at the extremes of the cascade can differ by several orders of magnitude at high Reynolds numbers. In between there is a range of scales (each one with its own characteristic length ) that has formed at the expense of the energy of the large ones. These scales are very large compared with the Kolmogorov length, but still very small compared with the large scale of the flow (i.e. ). Since eddies in this range are much larger than the dissipative eddies that exist at Kolmogorov scales, kinetic energy is essentially not dissipated in this range, and it is merely transferred to smaller scales until viscous effects become important as the order of the Kolmogorov scale is approached. Within this range inertial effects are still much larger than viscous effects, and it is possible to assume that viscosity does not play a role in their internal dynamics (for this reason this range is called "inertial range").

湍流拥有属性是一个层次的尺度，通过它能量级联发生。动能耗散发生在科尔莫戈罗夫长度级的尺度上，而能量输入到级联中来自大尺度的有序衰变。在高雷诺数下，这两个级联的极端尺度可以相差几百万数量级。在这两者之间有一系列的尺度(每个尺度都有自己的特征长度) ，这些尺度的形成是以大尺度的能量为代价的。这些尺度与科尔莫戈罗夫长度相比是非常大的，但是与流量的大尺度相比仍然是非常小的。由于在这个范围内的涡旋比在科尔莫戈罗夫尺度上存在的耗散涡旋要大得多，直到与科尔莫戈罗夫尺度的顺序接近时粘性效应开始变得重要，动能在这个范围内基本上没有消散，它只是转移到较小的尺度上。在这个范围内，惯性效应仍然比粘性效应大得多，可以假定粘性在其内部动力学中不起作用(因此这个范围称为”惯性范围”)。

Hence, a third hypothesis of Kolmogorov was that at very high Reynolds number the statistics of scales in the range *η* ≪ *r* ≪ *L* are universally and uniquely determined by the scale r and the rate of energy dissipation ε.

Hence, a third hypothesis of Kolmogorov was that at very high Reynolds number the statistics of scales in the range are universally and uniquely determined by the scale and the rate of energy dissipation .

因此，科尔莫哥罗夫的第三个假设是，在非常大的雷诺数范围内尺度的统计普遍且仅由尺度和能量耗散率决定的。

The way in which the kinetic energy is distributed over the multiplicity of scales is a fundamental characterization of a turbulent flow. For homogeneous turbulence (i.e., statistically invariant under translations of the reference frame) this is usually done by means of the *energy spectrum function* *E*(*k*), where k is the modulus of the wavevector corresponding to some harmonics in a Fourier representation of the flow velocity field **u**(**x**):

The way in which the kinetic energy is distributed over the multiplicity of scales is a fundamental characterization of a turbulent flow. For homogeneous turbulence (i.e., statistically invariant under translations of the reference frame) this is usually done by means of the energy spectrum function , where is the modulus of the wavevector corresponding to some harmonics in a Fourier representation of the flow velocity field :

动能在多重尺度上的分布方式是湍流的基本角色塑造。对于均匀湍流(即在参考系平移下具有统计不变性) ，这通常是通过能量谱函数来实现的，其中波矢的模量对应于流速场的傅里叶表示法中的某些谐波:

- [math]\displaystyle{ \mathbf{u}(\mathbf{x}) = \iiint_{\mathbb{R}^3} \hat{\mathbf{u}}(\mathbf{k})e^{i \mathbf{k \cdot x}} \, \mathrm{d}^3\mathbf{k} \,, }[/math]

[math]\displaystyle{ \mathbf{u}(\mathbf{x}) = \iiint_{\mathbb{R}^3} \hat{\mathbf{u}}(\mathbf{k})e^{i \mathbf{k \cdot x}} \, \mathrm{d}^3\mathbf{k} \,, }[/math]

{u}(\mathbf{x}) = \iiint_{\mathbb{R}^3} \hat{\mathbf{u}}(\mathbf{k})e^{i \mathbf{k \cdot x}} \,

where **û**(**k**) is the Fourier transform of the flow velocity field. Thus, *E*(*k*) d*k* represents the contribution to the kinetic energy from all the Fourier modes with *k* < 模板:Abs < *k* + d*k*, and therefore,

where is the Fourier transform of the flow velocity field. Thus, represents the contribution to the kinetic energy from all the Fourier modes with < k + dk}}, and therefore,

其中包含流速场的傅里叶变换。因此，用 k + dk 表示所有傅里叶模式对动能的贡献，因此,

- [math]\displaystyle{ \tfrac12\left\langle u_i u_i \right\rangle = \int_0^\infty E(k) \, \mathrm{d}k \,, }[/math]

[math]\displaystyle{ \tfrac12\left\langle u_i u_i \right\rangle = \int_0^\infty E(k) \, \mathrm{d}k \,, }[/math]

\tfrac12\left\langle u_i u_i \right\rangle = \int_0^\infty E(k) \,

where 模板:Sfrac⟨*u _{i}u_{i}*⟩ is the mean turbulent kinetic energy of the flow. The wavenumber k corresponding to length scale r is

*k*= 模板:Sfrac. Therefore, by dimensional analysis, the only possible form for the energy spectrum function according with the third Kolmogorov's hypothesis is

where ⟨u_{i}u_{i}⟩}} is the mean turbulent kinetic energy of the flow. The wavenumber corresponding to length scale is r is *k* = 模板:Sfrac. Therefore, by dimensional analysis, the only possible form for the energy spectrum function according with the third Kolmogorov's hypothesis is

其中⟨u_{i}u_{i}⟩是流动的平均湍动能。对应于长度尺度的波数是*k* = 模板:Sfrac value is very small, which explain the success of Kolmogorov theory in regards to low order statistical moments. In particular, it can be shown that when the energy spectrum follows a power law

有相当多的证据表明，紊流偏离了这种行为。标度指数偏离理论预测的n值，成为结构函数阶数的非线性函数。这些常数的普遍性也受到了质疑。对于低阶，与n值的差异很小，这解释了科尔莫戈罗夫理论在低阶统计矩方面的成功。特别地，我们可以证明，当能谱遵循幂定律时

- [math]\displaystyle{ E(k) \propto k^{-p} \,, }[/math]

[math]\displaystyle{ E(k) \propto k^{-p} \,, }[/math]

E(k) \propto k^{-p}

with 1 < *p* < 3, the second order structure function has also a power law, with the form

with , the second order structure function has also a power law, with the form

二阶结构函数也有幂律，与形式

- [math]\displaystyle{ \Big\langle \big (\delta \mathbf{u}(r)\big )^2 \Big\rangle \propto r^{p-1} \,, }[/math]

[math]\displaystyle{ \Big\langle \big (\delta \mathbf{u}(r)\big )^2 \Big\rangle \propto r^{p-1} \,, }[/math]

mathbf{u}(r)\big )^2 \Big\rangle \propto r^{p-1}

Since the experimental values obtained for the second order structure function only deviate slightly from the 模板:Sfrac value predicted by Kolmogorov theory, the value for p is very near to 模板:Sfrac (differences are about 2%^{[21]}). Thus the "Kolmogorov −模板:Sfrac spectrum" is generally observed in turbulence. However, for high order structure functions the difference with the Kolmogorov scaling is significant, and the breakdown of the statistical self-similarity is clear. This behavior, and the lack of universality of the C_{n} constants, are related with the phenomenon of intermittency in turbulence. This is an important area of research in this field, and a major goal of the modern theory of turbulence is to understand what is really universal in the inertial range.

Since the experimental values obtained for the second order structure function only deviate slightly from the value predicted by Kolmogorov theory, the value for is very near to (differences are about 2%). Thus the "Kolmogorov − spectrum" is generally observed in turbulence. However, for high order structure functions the difference with the Kolmogorov scaling is significant, and the breakdown of the statistical self-similarity is clear. This behavior, and the lack of universality of the constants, are related with the phenomenon of intermittency in turbulence. This is an important area of research in this field, and a major goal of the modern theory of turbulence is to understand what is really universal in the inertial range.

由于二阶结构函数的实验值与柯尔莫哥罗夫理论预测值只有很小的偏差，因此二阶结构函数的实验值与柯尔莫哥罗夫理论预测值非常接近(差异约为2%)。因此，在湍流中普遍观察到“科尔莫戈罗夫谱”。然而，对于高阶结构函数，柯尔莫哥罗夫标度的差异是显著的，统计自相似性的崩溃是明显的。这种现象和常数缺乏普适性与湍流中的间歇现象有关。这是这个领域的一个重要研究领域，现代湍流理论的一个主要目标是理解在惯性范围内什么是真正普遍的。

## See also

参见

## References and notes

参考资料和注释

- ↑ Batchelor, G. (2000).
*Introduction to Fluid Mechanics*. - ↑ Ting, F. C. K.; Kirby, J. T. (1996). "Dynamics of surf-zone turbulence in a spilling breaker".
*Coastal Engineering*.**27**(3–4): 131–160. doi:10.1016/0378-3839(95)00037-2. - ↑
^{3.0}^{3.1}Tennekes, H.; Lumley, J. L. (1972).*A First Course in Turbulence*. MIT Press. https://mitpress.mit.edu/books/first-course-turbulence. - ↑ Eames, I.; Flor, J. B. (January 17, 2011). "New developments in understanding interfacial processes in turbulent flows".
*Philosophical Transactions of the Royal Society A*.**369**(1937): 702–705. Bibcode:2011RSPTA.369..702E. doi:10.1098/rsta.2010.0332. PMID 21242127. - ↑ Kunze, Eric; Dower, John F.; Beveridge, Ian; Dewey, Richard; Bartlett, Kevin P. (2006-09-22). "Observations of Biologically Generated Turbulence in a Coastal Inlet".
*Science*(in English).**313**(5794): 1768–1770. Bibcode:2006Sci...313.1768K. doi:10.1126/science.1129378. ISSN 0036-8075. PMID 16990545. - ↑ Narasimha, R.; Rudra Kumar, S.; Prabhu, A.; Kailas, S. V. (2007). "Turbulent flux events in a nearly neutral atmospheric boundary layer" (PDF).
*Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences*.**365**(1852): 841–858. Bibcode:2007RSPTA.365..841N. doi:10.1098/rsta.2006.1949. PMID 17244581. - ↑ Trevethan, M.; Chanson, H. (2010). "Turbulence and Turbulent Flux Events in a Small Estuary".
*Environmental Fluid Mechanics*.**10**(3): 345–368. doi:10.1007/s10652-009-9134-7. - ↑ Jin, Y.; Uth, M.-F.; Kuznetsov, A. V.; Herwig, H. (2 February 2015). "Numerical investigation of the possibility of macroscopic turbulence in porous media: a direct numerical simulation study".
*Journal of Fluid Mechanics*.**766**: 76–103. Bibcode:2015JFM...766...76J. doi:10.1017/jfm.2015.9. - ↑ Ferziger, Joel H.; Peric, Milovan (2002).
*Computational Methods for Fluid Dynamics*. Germany: Springer-Verlag Berlin Heidelberg. pp. 265–307. . - ↑ Kundu, Pijush K.; Cohen, Ira M.; Dowling, David R. (2012).
*Fluid Mechanics*. Netherlands: Elsevier Inc. pp. 537–601. . - ↑ Kundu, Pijush K.; Cohen, Ira M.; Dowling, David R. (2012).
*Fluid Mechanics*. Netherlands: Elsevier Inc. pp. 537–601. . - ↑ Kundu, Pijush K.; Cohen, Ira M.; Dowling, David R. (2012).
*Fluid Mechanics*. Netherlands: Elsevier Inc. pp. 537–601. . - ↑
^{13.0}^{13.1}Tennekes, Hendrik (1972).*A First Course in Turbulence*. The MIT Press. - ↑ weizmann.ac.il
- ↑ Marshak, Alex (2005).
*3D radiative transfer in cloudy atmospheres*. Springer. p. 76. ISBN 978-3-540-23958-1. https://books.google.com/books?id=wzg6wnpHyCUC. - ↑ Mullin, Tom (11 November 1989). "Turbulent times for fluids".
*New Scientist*. - ↑ Davidson, P. A. (2004).
*Turbulence: An Introduction for Scientists and Engineers*. Oxford University Press. ISBN 978-0-19-852949-1. https://books.google.com/?id=rkOmKzujZB4C&pg=PA24&dq=%22when+I+die+and+go+to+Heaven+there+are+two+matters+on+which+I+hope+for+enlightenment%22#v=onepage&q=%22when%20I%20die%20and%20go%20to%20Heaven%20there%20are%20two%20matters%20on%20which%20I%20hope%20for%20enlightenment%22&f=false. - ↑ Falkovich, G. (2011).
*Fluid Mechanics*. Cambridge University Press.模板:Missing ISBN - ↑ Sommerfeld, Arnold (1908). "Ein Beitrag zur hydrodynamischen Erkläerung der turbulenten Flüssigkeitsbewegüngen" [A Contribution to Hydrodynamic Explanation of Turbulent Fluid Motions].
*International Congress of Mathematicians*.**3**: 116–124. - ↑ Avila, K.; Moxey, D.; de Lozar, A.; Avila, M.; Barkley, D.; B. Hof (July 2011). "The Onset of Turbulence in Pipe Flow".
*Science*.**333**(6039): 192–196. Bibcode:2011Sci...333..192A. doi:10.1126/science.1203223. PMID 21737736. - ↑ Mathieu, J.; Scott, J. (2000).
*An Introduction to Turbulent Flow*. Cambridge University Press.模板:Missing ISBN

## Further reading

进一步阅读

### General

通用

- Falkovich, Gregory; Sreenivasan, K. R. (April 2006). "Lessons from hydrodynamic turbulence" (PDF).
*Physics Today*.**59**(4): 43–49. Bibcode:2006PhT....59d..43F. doi:10.1063/1.2207037.

- Frisch, U. (1995).
*Turbulence: The Legacy of A. N. Kolmogorov*. Cambridge University Press. ISBN 9780521457132.

- Davidson, P. A. (2004).
*Turbulence – An Introduction for Scientists and Engineers*. Oxford University Press. ISBN 978-0198529491.

- Cardy, J.; Falkovich, G.; Gawedzki, K. (2008).
*Non-equilibrium Statistical Mechanics and Turbulence*. Cambridge University Press. ISBN 9780521715140.

- Durbin, P. A.; Pettersson Reif, B. A. (2001).
*Statistical Theory and Modeling for Turbulent Flows*. John Wiley & Sons. ISBN 978-0-470-68931-8.

- Bohr, T.; Jensen, M. H.; Paladin, G.; Vulpiani, A. (1998).
*Dynamical Systems Approach to Turbulence*. Cambridge University Press. ISBN 9780521475143.

- McDonough, J. M. (2007). "Introductory Lectures on Turbulence – Physics, Mathematics, and Modeling" (PDF). University of Kentucky.

- Nieuwstadt, F. T. M.; Boersma, B. J.; Westerweel, J. (2016).
*Turbulence – Introduction to Theory and Applications of Turbulent Flows*(Online ed.). Springer. ISBN 978-3-319-31599-7.

### Original scientific research papers and classic monographs

原创科研论文和经典专著

- Kolmogorov, Andrey Nikolaevich (1941). "The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers".
*Proceedings of the USSR Academy of Sciences*(in русский).**30**: 299–303.

- Translated into English: Kolmogorov, Andrey Nikolaevich (July 8, 1991). Translated by Levin, V. "The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers" (PDF).
*Proceedings of the Royal Society A*.**434**(1991): 9–13. Bibcode:1991RSPSA.434....9K. doi:10.1098/rspa.1991.0075. Archived from the original (PDF) on September 23, 2015.

- Translated into English: Kolmogorov, Andrey Nikolaevich (July 8, 1991). Translated by Levin, V. "The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers" (PDF).

- Kolmogorov, Andrey Nikolaevich (1941). "Dissipation of Energy in Locally Isotropic Turbulence".
*Proceedings of the USSR Academy of Sciences*(in русский).**32**: 16–18.

- Translated into English: Kolmogorov, Andrey Nikolaevich (July 8, 1991). "Dissipation of energy in the locally isotropic turbulence" (PDF).
*Proceedings of the Royal Society A*.**434**(1980): 15–17. Bibcode:1991RSPSA.434...15K. doi:10.1098/rspa.1991.0076. Archived from the original (PDF) on July 6, 2011.

- Translated into English: Kolmogorov, Andrey Nikolaevich (July 8, 1991). "Dissipation of energy in the locally isotropic turbulence" (PDF).

- Batchelor, G. K. (1953).
*The Theory of Homogeneous Turbulence*. Cambridge University Press.

## External links

外部链接

- Center for Turbulence Research, Scientific papers and books on turbulence

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