更改

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

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

* Smoke rising from a [[cigarette]]. For the first few centimeters, the smoke is [[Laminar flow|laminar]]. The smoke [[Plume (fluid dynamics)|plume]] becomes turbulent as its [[Reynolds number]] increases with increases in flow velocity and characteristic lengthscale.

* Smoke rising from a [[cigarette]]. For the first few centimeters, the smoke is [[Laminar flow|laminar]]. The smoke [[Plume (fluid dynamics)|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。最开始的几厘米，烟雾是层流。'''<font color="#ff8000"> 雷诺数Reynolds number </font>'''随着流速和特征长度增加时，飘起来的烟变成了湍流。

* 烟雾从一支香烟上升起。最开始的几厘米，烟雾是层流。'''<font color="#ff8000"> 雷诺数Reynolds number </font>'''随着流速和特征长度增加时，飘起来的烟变成了湍流。

* 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.

* 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.

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

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

飞机飞行时经视宁历过'''<font color="#ff8000"> 晴空湍流Clear-air turbulence</font>'''，以及'''<font color="#ff8000"> 天文视宁度astronomical seeing</font>'''不佳（通过大气看到的图像模糊）。

 

* 飞机飞行时经视宁历过'''<font color="#ff8000"> 晴空湍流Clear-air turbulence</font>'''，以及'''<font color="#ff8000"> 天文视宁度astronomical seeing</font>'''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

* 大部分陆地的大气环流

* The oceanic and atmospheric [[mixed layer]]s and intense oceanic currents.

* The oceanic and atmospheric [[mixed layer]]s and intense oceanic currents.

* 海洋和大气混合层和强烈的洋流

* 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 [[scrubber]]s, [[dynamic scraped surface heat exchanger]]s, etc.) and machines (for instance, [[internal combustion engine]]s and [[gas turbine]]s).

* The flow conditions in many industrial equipment (such as pipes, ducts, precipitators, gas [[scrubber]]s, [[dynamic scraped surface heat exchanger]]s, etc.) and machines (for instance, [[internal combustion engine]]s and [[gas turbine]]s).

* 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'''<font color="#ff8000"> 内燃机internal combustion engine</font>'''、'''<font color="#ff8000"> 燃气轮机gas turbine </font>''')中的流动状况。

* 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'''<font color="#ff8000"> 内燃机internal combustion engine</font>'''、'''<font color="#ff8000"> 燃气轮机gas turbine </font>''')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。

* 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。

* 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.

* Biologically generated turbulence resulting from swimming animals affects ocean mixing. <ref>{{Cite journal|title = Observations of Biologically Generated Turbulence in a Coastal Inlet|url = http://science.sciencemag.org/content/313/5794/1768|journal = Science|date = 2006-09-22|issn = 0036-8075|pmid = 16990545|pages = 1768–1770|volume = 313|issue = 5794|doi = 10.1126/science.1129378|language = en|first = Eric|last = Kunze|first2 = John F.|last2 = Dower|first3 = Ian|last3 = Beveridge|first4 = Richard|last4 = Dewey|first5 = Kevin P.|last5 = Bartlett|bibcode = 2006Sci...313.1768K }}</ref>

* Biologically generated turbulence resulting from swimming animals affects ocean mixing. <ref>{{Cite journal|title = Observations of Biologically Generated Turbulence in a Coastal Inlet|url = http://science.sciencemag.org/content/313/5794/1768|journal = Science|date = 2006-09-22|issn = 0036-8075|pmid = 16990545|pages = 1768–1770|volume = 313|issue = 5794|doi = 10.1126/science.1129378|language = en|first = Eric|last = Kunze|first2 = John F.|last2 = Dower|first3 = Ian|last3 = Beveridge|first4 = Richard|last4 = Dewey|first5 = Kevin P.|last5 = Bartlett|bibcode = 2006Sci...313.1768K }}</ref>

* 游泳动物引起的生物湍流会影响海洋混合。

* 游泳动物引起的生物湍流会影响海洋混合。

* [[Snow fence]]s work by inducing turbulence in the wind, forcing it to drop much of its snow load near the fence.

* [[Snow fence]]s 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。

* 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.

* 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.<ref name="Narasimha">{{cite journal|last1=Narasimha|first1=R.|last2=Rudra Kumar|first2=S.|last3=Prabhu|first3=A.|last4=Kailas|first4=S. V.|title= Turbulent flux events in a nearly neutral atmospheric boundary layer|journal=Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences|volume=365|issue=1852|pages=841–858 |year=2007 | doi=10.1098/rsta.2006.1949|pmid=17244581|bibcode = 2007RSPTA.365..841N |url=http://repository.ias.ac.in/24526/1/322.pdf}}</ref><ref name="Trevethan_Chanson">{{cite journal|last1=Trevethan|first1=M.|author2-link=Hubert Chanson|last2=Chanson|first2=H.|title= Turbulence and Turbulent Flux Events in a Small Estuary|journal=[[Environmental Fluid Mechanics]]|issue=3|volume=10 |pages=345–368 |year=2010 |isbn= | doi=10.1007/s10652-009-9134-7 |url=http://espace.library.uq.edu.au/view/UQ:205133}}</ref> 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.<ref name="Narasimha">{{cite journal|last1=Narasimha|first1=R.|last2=Rudra Kumar|first2=S.|last3=Prabhu|first3=A.|last4=Kailas|first4=S. V.|title= Turbulent flux events in a nearly neutral atmospheric boundary layer|journal=Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences|volume=365|issue=1852|pages=841–858 |year=2007 | doi=10.1098/rsta.2006.1949|pmid=17244581|bibcode = 2007RSPTA.365..841N |url=http://repository.ias.ac.in/24526/1/322.pdf}}</ref><ref name="Trevethan_Chanson">{{cite journal|last1=Trevethan|first1=M.|author2-link=Hubert Chanson|last2=Chanson|first2=H.|title= Turbulence and Turbulent Flux Events in a Small Estuary|journal=[[Environmental Fluid Mechanics]]|issue=3|volume=10 |pages=345–368 |year=2010 |isbn= | doi=10.1007/s10652-009-9134-7 |url=http://espace.library.uq.edu.au/view/UQ:205133}}</ref> 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.

* 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。

* 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.

* Recently, turbulence in porous media became a highly debated subject.<ref>{{cite journal|last1=Jin|first1=Y.|last2=Uth|first2=M.-F.|last3=Kuznetsov|first3=A. V.|last4=Herwig|first4=H.|title=Numerical investigation of the possibility of macroscopic turbulence in porous media: a direct numerical simulation study|journal=Journal of Fluid Mechanics|date=2 February 2015|volume=766|pages=76–103|doi=10.1017/jfm.2015.9|bibcode = 2015JFM...766...76J }}</ref>

* Recently, turbulence in porous media became a highly debated subject.<ref>{{cite journal|last1=Jin|first1=Y.|last2=Uth|first2=M.-F.|last3=Kuznetsov|first3=A. V.|last4=Herwig|first4=H.|title=Numerical investigation of the possibility of macroscopic turbulence in porous media: a direct numerical simulation study|journal=Journal of Fluid Mechanics|date=2 February 2015|volume=766|pages=76–103|doi=10.1017/jfm.2015.9|bibcode = 2015JFM...766...76J }}</ref>

* 最近，多孔介质中的湍流成为一个备受争议的话题

* 最近，多孔介质中的湍流成为一个备受争议的话题

  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，并非所有的混乱流都是如此。

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。

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，即拥有属性涡旋伸展。在流体动力学中，它们本质上是受拉伸作用的涡旋，与拉伸方向相应的涡量分量增加有关---- 由于角动量守恒定律。另一方面，涡旋伸展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，以至于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，但是它们基本上二维的形状不允许涡旋的产生，因此不会产生湍流。另一方面，海洋流动是分散的，但本质上是非旋转的，因此不是湍流的。

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。

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: 为了维持湍流，需要一个持久的能量供应源，因为当动能通过粘性剪切应力转化为内能时，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，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。

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'''<font color="#ff8000"> 粘性切应力viscous shear stress </font>'''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，湍流可以实现为一个平均流上的流速起伏和涡流谱的叠加。这些涡流被粗略地定义为流速、涡量和压力的相干型。湍流可以被看作是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和高度不对称的。然而，根据这些长度尺度，这些涡旋可以分为三类。

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

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

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

数学 t 左( frac { langle u‘ u’ rangle 右) 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&prime; is the velocity fluctuation, and <math>\tau</math> is the time lag between measurements.

where u&prime; is the velocity fluctuation, and <math>\tau</math> is the time lag between measurements.

其中 u & prime 是速度波动，而 math  tau / math 是两次测量之间的时间差。

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  

  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

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

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

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

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

数学 l 左(frac {1}{1} 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''&prime; is the velocity fluctuation in that same direction.<ref name="Tennekes 1972"/>

: where ''r'' is the distance between two measurement locations, and ''u''&prime; is the velocity fluctuation in that same direction.<ref name="Tennekes 1972"/>