# 数学代写|应用数学代写applied mathematics代考|Math2090

## 数学代写|应用数学代写applied mathematics代考|The Kolmogorov length scale

The only length scale that can be constructed from the dissipation rate $\epsilon$ and the kinematic viscosity $\nu$, called the Kolmogorov length scale, is
$$\eta=\left(\frac{\nu^3}{\epsilon}\right)^{1 / 4} \text {. }$$
The K41 theory implies that the dissipation length scale is of the order $\eta$.
If the energy dissipation rate is the same at all length scales, then, neglecting order one factors, we have
$$\epsilon=\frac{U^3}{L}$$
where $L, U$ are the integral length and velocity scales. Denoting by $\mathrm{R}_L$ the Reynolds number based on these scales,
$$\mathrm{R}_L=\frac{U L}{\nu},$$
it follows that
$$\frac{L}{\eta}=\mathrm{R}_L^{3 / 4} .$$
Thus, according to this dimensional analysis, the ratio of the largest (integral) length scale and the smallest (dissipation) length scale grows like $\mathrm{R}_L^{3 / 4}$ as $\mathrm{R}_L \rightarrow \infty$.
In order to resolve the finest length scales of a three-dimensional flow with integral-scale Reynolds number $\mathrm{R}_L$, we therefore need on the order of
$$N_L=\mathrm{R}_L^{9 / 4}$$
independent degrees of freedom (for example, $N_L$ Fourier coefficients of the velocity components). The rapid growth of $N_L$ with $\mathrm{R}_L$ limits the Reynolds numbers that can be attained in direct numerical simulations of turbulent flows.

## 数学代写|应用数学代写applied mathematics代考|Validity of the five-thirds law

Experimental observations, such as those made by Grant, Stewart and Moilliet (1962) in a tidal channel between islands off Vancouver, agree well with the fivethirds law for the energy spectrum, and give $C \approx 1.5$ in (2.27). The results of DNS on periodic ‘boxes’, using up to $4096^3$ grid points, are also in reasonable agreement with this prediction.

Although the energy spectrum predicted by the K41 theory is close to what is observed, there is evidence that it is not exactly correct. This would imply that there is something wrong with its original assumptions.

Kolmogorov and Oboukhov proposed a refinement of Kolmogorov’s original theory in 1962. It is, in particular, not clear that the energy dissipation rate $\epsilon$ should be assumed constant, since the energy dissipation in a turbulent flow itself varies over multiple length scales in a complicated fashion. This phenomenon, called ‘intermittency,’ can lead to corrections in the five-thirds law [23]. All such turbulence theories, however, depend on some kind of initial assumptions whose validity can only be checked by comparing their predictions with experimental or numerical observations.

As the above examples from fluid mechanics illustrate, dimensional arguments can lead to surprisingly powerful results, even without a detailed analysis of the underlying equations. All that is required is a knowledge of the quantities on which the problem being studied depends together with their dimensions. This does mean, however, one has to know the basic laws that govern the problem, and the dimensional constants they involve. Thus, contrary to the way it sometimes appears, dimensional analysis does not give something for nothing; it can only give what is put in from the start.

This fact cuts both ways. Many of the successes of dimensional analysis, such as Kolmogorov’s theory of turbulence, are the result of an insight into which dimensional parameters play an crucial role in a problem and which parameters can be ignored. Such insights typical depend upon a great deal of intuition and experience, and they may be difficult to justify or prove. 2

Conversely, it may happen that some dimensional parameters that appear to be so small they can be neglected have a significant effect, in which case scaling laws derived on the basis of dimensional arguments that ignore them are likely to be incorrect.

# 应用数学代考

## 数学代写|应用数学代写applied mathematics代考|The Kolmogorov length scale

$$\eta=\left(\frac{\nu^3}{\epsilon}\right)^{1 / 4} .$$
K41 理论意味着耗散长度标度为 $\eta$.

$$\epsilon=\frac{U^3}{L}$$

$$\mathrm{R}_L=\frac{U L}{\nu},$$

$$\frac{L}{\eta}=\mathrm{R}_L^{3 / 4} .$$

$$N_L=\mathrm{R}_L^{9 / 4}$$

## 数学代写|应用数学代写applied mathematics代考|Validity of the five-thirds law

Kolmogorov 和 Oboukhov 在 1962 年提出了对 Kolmogorov 原始理论的改进。ε应该假定为常数，因为湍流本身的能量耗散在多个长度尺度上以复杂的方式变化。这种称为“间歇性”的现象可以导致三分之五定律的修正 [23]。然而，所有这些湍流理论都依赖于某种初始假设，其有效性只能通过将它们的预测与实验或数值观察进行比较来检验。

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