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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").

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(即。).由于在这个范围内的涡旋比在 Kolmogorov 尺度上存在的耗散涡旋要大得多，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，直到粘性效应变得重要，因为科尔莫戈罗夫尺度的顺序接近。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”)。

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  are universally and uniquely determined by the scale  and the rate of energy dissipation .

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 .

<math>\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>\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}} \,

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

 

where {{math|'''û'''('''k''')}} is the Fourier transform of the flow velocity field. Thus, {{math|''E''(''k'')&nbsp;d''k''}} represents the contribution to the kinetic energy from all the Fourier modes with {{math|''k'' < {{abs|'''k'''}} < ''k'' + d''k''}}, and therefore,

where {{math|'''û'''('''k''')}} is the Fourier transform of the flow velocity field. Thus, {{math|''E''(''k'')&nbsp;d''k''}} represents the contribution to the kinetic energy from all the Fourier modes with {{math|''k'' < {{abs|'''k'''}} < ''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,

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,

流速场的傅里叶变换在哪里。因此，用 k + dk }表示所有傅里叶模式对动能的贡献，因此,

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

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

 

 

 

where {{math|{{sfrac|1|2}}⟨''u_iu_i''⟩}} is the mean turbulent kinetic energy of the flow. The wavenumber {{mvar|k}} corresponding to length scale {{mvar|r}} is {{math|''k'' {{=}} {{sfrac|2π|''r''}}}}. Therefore, by dimensional analysis, the only possible form for the energy spectrum function according with the third Kolmogorov's hypothesis is

where {{math|{{sfrac|1|2}}⟨''u_iu_i''⟩}} is the mean turbulent kinetic energy of the flow. The wavenumber {{mvar|k}} corresponding to length scale {{mvar|r}} is {{math|''k'' {{=}} {{sfrac|2π|''r''}}}}. Therefore, by dimensional analysis, the only possible form for the energy spectrum function according with the third Kolmogorov's hypothesis is

where ⟨u_iu_i⟩}} is the mean turbulent kinetic energy of the flow. The wavenumber  corresponding to length scale  is }}. Therefore, by dimensional analysis, the only possible form for the energy spectrum function according with the third Kolmogorov's hypothesis is

where ⟨u_iu_i⟩}} is the mean turbulent kinetic energy of the flow. The wavenumber  corresponding to length scale  is {{mvar|r}} is {{math|''k'' {{=}} {{sfrac|2π|''r''}}}}. Therefore, by dimensional analysis, the only possible form for the energy spectrum function according with the third Kolmogorov's hypothesis is

其中⟨u_iu_i⟩是流动的平均湍动能。对应于长度尺度的波数是}。因此，到了量纲分析，符合 Kolmogorov 第三假说的能谱函数唯一可能的形式是

其中⟨u_iu_i⟩是流动的平均湍动能。对应于长度尺度的波数是''k'' {{=}} {{sfrac|2π|''r''。因此，到了量纲分析，符合科尔莫哥罗夫第三假说的能谱函数唯一可能的形式是

where  would be a universal constant. This is one of the most famous results of Kolmogorov 1941 theory, and considerable experimental evidence has accumulated that supports it.

where  would be a universal constant. This is one of the most famous results of Kolmogorov 1941 theory, and considerable experimental evidence has accumulated that supports it.

In spite of this success, Kolmogorov theory is at present under revision. This theory implicitly assumes that the turbulence is statistically self-similar at different scales. This essentially means that the statistics are scale-invariant in the inertial range. A usual way of studying turbulent flow velocity fields is by means of flow velocity increments:

In spite of this success, Kolmogorov theory is at present under revision. This theory implicitly assumes that the turbulence is statistically self-similar at different scales. This essentially means that the statistics are scale-invariant in the inertial range. A usual way of studying turbulent flow velocity fields is by means of flow velocity increments:

尽管取得了这样的成功，科尔莫戈罗夫理论目前正在修正之中。这个理论隐含地假设了湍流在不同尺度上在统计上是自相似的。这实质上意味着统计量在惯性范围内是尺度不变的。研究湍流速度场的一种常用方法是用流速增量:

尽管取得了这样的成功，科尔莫戈罗夫理论目前仍在修正之中。这个理论隐含地假设了湍流在不同尺度上在统计上是自相似的。这实质上意味着统计量在惯性范围内是尺度不变的。研究湍流速度场的一种常用方法是用流速增量:

that is, the difference in flow velocity between points separated by a vector {{math|'''r'''}} (since the turbulence is assumed isotropic, the flow velocity increment depends only on the modulus of {{math|'''r'''}}). Flow velocity increments are useful because they emphasize the effects of scales of the order of the separation {{mvar|r}} when statistics are computed.  The statistical scale-invariance implies that the scaling of flow velocity increments should occur with a unique scaling exponent {{mvar|β}}, so that when {{mvar|r}} is scaled by a factor {{mvar|λ}},

that is, the difference in flow velocity between points separated by a vector {{math|'''r'''}} (since the turbulence is assumed isotropic, the flow velocity increment depends only on the modulus of {{math|'''r'''}}). Flow velocity increments are useful because they emphasize the effects of scales of the order of the separation {{mvar|r}} when statistics are computed.  The statistical scale-invariance implies that the scaling of flow velocity increments should occur with a unique scaling exponent {{mvar|β}}, so that when {{mvar|r}} is scaled by a factor {{mvar|λ}},

that is, the difference in flow velocity between points separated by a vector  (since the turbulence is assumed isotropic, the flow velocity increment depends only on the modulus of ). Flow velocity increments are useful because they emphasize the effects of scales of the order of the separation  when statistics are computed.  The statistical scale-invariance implies that the scaling of flow velocity increments should occur with a unique scaling exponent , so that when  is scaled by a factor ,

that is, the difference in flow velocity between points separated by a vector  (since the turbulence is assumed isotropic, the flow velocity increment depends only on the modulus of '''r'''). Flow velocity increments are useful because they emphasize the effects of scales of the order of the separation  when statistics are computed.  The statistical scale-invariance implies that the scaling of flow velocity increments should occur with a unique scaling exponent , so that when  is scaled by a factor ,

也就是说，由矢量分隔的点之间的流速差(由于湍流假定为各向同性，流速增量只取决于。流速增量是有用的，因为它们在计算统计量时强调了分离顺序尺度的影响。统计标度不变性意味着流速增量的标度应该以唯一的标度指数发生，因此当被一个因子标度时,

也就是说，由矢量分隔的点之间的流速差(由于湍流假定为各向同性，流速增量只取决于r。流速增量是有用的，因为它们在计算统计量时强调了分离顺序尺度的影响。统计标度不变性意味着流速增量的标度应该以唯一的标度指数发生，因此当被一个因子标度时,

with  independent of the scale . From this fact, and other results of Kolmogorov 1941 theory, it follows that the statistical moments of the flow velocity increments (known as structure functions in turbulence) should scale as

with  independent of the scale . From this fact, and other results of Kolmogorov 1941 theory, it follows that the statistical moments of the flow velocity increments (known as structure functions in turbulence) should scale as

与音阶无关。根据这一事实，以及 Kolmogorov 1941理论的其他结果，得出流速增量的统计矩(称为紊流中的结构函数)应该按照

与音阶无关。根据这一事实，以及科尔莫戈罗夫1941理论的其他结果，得出流速增量的统计矩(称为紊流中的结构函数)应该按照

<math>\Big\langle \big ( \delta \mathbf{u}(r)\big )^n \Big\rangle = C_n (\varepsilon r)^\frac{n}{3} \,,</math>

<math>\Big\langle \big ( \delta \mathbf{u}(r)\big )^n \Big\rangle = C_n (\varepsilon r)^\frac{n}{3} \,,</math>

数学大角度大角度(δ (u)(r) Big) ^ n 大角度 c n (varepsilon r) ^ frac { n }3，/ math

数学大角度大角度(δ (u)(r) Big) ^ n 大角度 c n (varepsilon r) ^ frac { n }3，

There is considerable evidence that turbulent flows deviate from this behavior. The scaling exponents deviate from the {{math|{{sfrac|''n''|3}}}} value predicted by the theory, becoming a non-linear function of the order {{mvar|n}} of the structure function. The universality of the constants have also been questioned. For low orders the discrepancy with the Kolmogorov {{math|{{sfrac|''n''|3}}}} 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

There is considerable evidence that turbulent flows deviate from this behavior. The scaling exponents deviate from the {{math|{{sfrac|''n''|3}}}} value predicted by the theory, becoming a non-linear function of the order {{mvar|n}} of the structure function. The universality of the constants have also been questioned. For low orders the discrepancy with the Kolmogorov {{math|{{sfrac|''n''|3}}}} 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

There is considerable evidence that turbulent flows deviate from this behavior. The scaling exponents deviate from the }} value predicted by the theory, becoming a non-linear function of the order  of the structure function. The universality of the constants have also been questioned. For low orders the discrepancy with the Kolmogorov }} 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

There is considerable evidence that turbulent flows deviate from this behavior. The scaling exponents deviate from the{{math|{{sfrac|''n''|3}}}} value predicted by the theory, becoming a non-linear function of the order  of the structure function. The universality of the constants have also been questioned. For low orders the discrepancy with the Kolmogorov }} 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

with , 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

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

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

数学大角度大(delta 数学 u }(r) Big) ^ 2大角度前进到 r ^ { p-1} ，，/ 数学

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

 

===General===

===General===

* {{cite journal|first1=Gregory|last1=Falkovich|first2=K. R.|last2=Sreenivasan|title=Lessons from hydrodynamic turbulence|journal=[[Physics Today]]|volume=59|issue=4|pages=43–49|date=April 2006|url=http://www.phy.olemiss.edu/~jgladden/phys510/spring06/turbulence.pdf|doi=10.1063/1.2207037|bibcode=2006PhT....59d..43F}}

* {{cite journal|first1=Gregory|last1=Falkovich|first2=K. R.|last2=Sreenivasan|title=Lessons from hydrodynamic turbulence|journal=[[Physics Today]]|volume=59|issue=4|pages=43–49|date=April 2006|url=http://www.phy.olemiss.edu/~jgladden/phys510/spring06/turbulence.pdf|doi=10.1063/1.2207037|bibcode=2006PhT....59d..43F}}

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