# K41 Kolmogorov scaling relation and structure functions (turbulence)

• nicholasmr
In summary, the K41 scaling relation for homogeneous and isotropic turbulence is based on the assumption that the average velocity difference between two points is proportional to the mean energy dissipation and the length of displacement between the two points.
nicholasmr
Hi there.

I am having trouble interpreting the Kolmogorov K41 scaling relation for homogeneous and isotropic turbulence:

$S_{p}(l)$ = <$\delta u(l)^{p}$> = <$|u(r+l) - u(r)|^{p}$> $\propto$ ($\epsilon l$)$^{p/3}$

where $l$ is the length of displacement between two points under consideration in the flow, $\epsilon$ is the mean energy dissipation, and $u(r)$ is the velocity at point $r$ in the flow.

My question is:
Since we assume the flow to be homogeneous and isotropic in a statistically averaged sense (fully developed turbulence), how come <$\delta u(l)^{p}$> is not zero? If the turbulent flow is homogeneous and isotropic on average, then the (averaged) flow in the two point should be similar? I cannot see the $l$-dependence in my mental picture of this situation?

Regards Nicholas.

The Kolmogorov K41 scaling relation for homogeneous and isotropic turbulence does not assume that the flow between two points is similar. Instead, it assumes that the average velocity difference between two points is equal to the product of the mean energy dissipation and the length of displacement between the two points. This is due to the fact that turbulent flows are chaotic and unpredictable and so the velocity differences between two nearby points can vary greatly. This is why the K41 scaling relation is useful in predicting the behavior of homogeneous and isotropic turbulence.

## 1. What is the K41 Kolmogorov scaling relation?

The K41 Kolmogorov scaling relation is a theory proposed by Russian mathematician Andrey Kolmogorov in 1941 to explain the behavior of turbulence in fluid flows. It states that in fully developed turbulence, the energy cascade from large scales to small scales follows a universal power law, with the energy dissipation rate being proportional to the third power of the wavenumber.

## 2. How is the K41 Kolmogorov scaling relation related to structure functions?

The K41 Kolmogorov scaling relation is intimately connected to the concept of structure functions in turbulence. Structure functions measure the statistical properties of a scalar quantity (such as velocity or energy) at two different points in a turbulent flow. The K41 theory predicts that the scaling of these structure functions with separation distance will follow a specific power law, which has been confirmed by experimental and numerical studies.

## 3. What are the implications of the K41 Kolmogorov scaling relation?

The K41 Kolmogorov scaling relation has important implications for our understanding of turbulence and its applications. It provides a theoretical framework for predicting the statistical properties of turbulence, such as energy dissipation rates, and has been used to develop turbulence models for engineering applications.

## 4. Is the K41 Kolmogorov scaling relation universally applicable to all turbulent flows?

No, the K41 Kolmogorov scaling relation is a simplified model that only applies to fully developed turbulence in an idealized, homogeneous and isotropic flow. In reality, turbulent flows can exhibit different scaling behaviors depending on factors such as the flow geometry, boundary conditions, and energy input mechanisms.

## 5. How has the K41 Kolmogorov scaling relation been validated?

The K41 Kolmogorov scaling relation has been extensively validated through experiments and numerical simulations, with good agreement between the predicted and observed scaling behaviors. However, some deviations from the idealized K41 model have been observed, leading to the development of more complex theories such as the extended self-similarity hypothesis.

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