Andrey Kolmogorov famously wrote three terse papers in 1941 on turbulence. They were published in an English translation in 1991 by Royal Society Publishing. At p. 13 in the first paper and at p. 15 in the third he shows 4/3 scaling using first and second moments. He employs a function B with a double subscript in his derivation. What is the function B? What is the physical description corresponding to his mathematics? G. K. Batchelor in 1946 wrote on Kolmogoroff’s theory of locally isotropic turbulence mentions tensors and vectors but does not give an elementary interpretation of Kolmogorov’s derivation. Can you?
I don't think you'll find Richardson's 4/3 law explicitely derived in K41, however, you can derive it directly using some of the results from there: since your diffusivity depends on the velocity scale U associated with an eddy of size r, the velocity scale U can be obtained from B_dd (the second order structure function) or E_dd (the power spectral density), in any case, U is proportional to r^1/3 in the inertial range since B_dd~r^2/3 (or, equivalently E_dd~k^-5/3). You could do the same starting from C_\epsilon=\epsilon r / U^3=const and working your way back to a diffusivity coefficient.
Assuming you're talking about the 4/3 coefficients in eq. (24) in K41a and eq. (21) in K41c, these involve second order moments of the velocity structure function (not first and second) and this specific 4/3 appears purely a consequence of isotropy.
- What is the function B?
B is defined as a structure function, which in the context of K41 is just another measure for two-point correlations. Broadly speaking B is a real-space analog to the energy spectrum E. If E(k) characterises the energy associated with wavenumber k then B_dd(r) characterises the energy associated with a length scale r.
Take a homogenous, isotropic and statistically stationary flow and consider two points (say A and B) along a given axis (say x_d). If u is the velocity component in the direction x_d, then for every time instant/independent sample, you can compute (u_A - u_B)^p for some integer p, where u_A is the fluctuating velocity component at point A and u_B is the fluctuating velocity component at point B.
If you set p=2 and take the time/ensemble average () you obtain the second order structure function B_dd=, for p=3 you get the third order structure function B_ddd= and so on. The relevance of these quantities comes in the answer to your question below, but notice how these functions relate to the two-point correlation function: for example B_dd=2 -2 (because of homogeneity = = ); the analogy between the structure function and the correlation function means you can jump to E_dd by Fourier transforming B_dd/2- (see Wiener-Khinchin theorem).
Now consider v, the velocity component normal to x_d. We can define, as above, the structure functions B_nn=, B_nnn=, etc. B_nn and B_dd are related through isotropy (see e.g. https://arc.aiaa.org/doi/abs/10.2514/8.350).
Because K41 considers only homogenous, isotropic and statistically stationary flows, it suffices to consider a single velocity component in a single direction to fully describe the turbulence. For an assumption-free approach see https://arxiv.org/pdf/physics/0102063.pdf for structure functions and https://link.springer.com/article/10.1007/BF01329645 for energy spectra.
- What is the physical description corresponding to his mathematics?
In a crude way (see refs below), the velocity difference (u_A - u_B) is a characteristic velocity for eddies of size r_AB, with r_AB being the distance between points A and B. Thus, B_dd characterises the kinetic energy of eddies of size r_AB or smaller and B_ddd characterises the transport in the eddy-reference-frame of that kinetic energy.
Notice that these are analogous, but very different, to standard one-point measures for kinetic energy () and turbulent transport () as they include information about two-points (i.e. length scales) in the flow.
You can find extensive (and very good) discussions on the physical interpretation of these functions in turbulence text books such as Frisch's "Turbulence: The Legacy of A. N. Kolmogorov" and Davidson's "Turbulence: An Introduction for Scientists and Engineers". Or have a look at both Article Identifying Turbulent Energy Distributions in Real, Rather t...
and https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/on-the-deficiency-of-evenorder-structure-functions-as-inertialrange-diagnostics/20C72BC567F69524E1F25A1E6AEA19A5 which are very relevant to this question.One advantage of using structure functions as opposed to energy spectra is that they are somewhat more tenable/easy to define when dealing with inhomogeneous/anisotropic flows: for example, defining the spectrum near a solid boundary can be tricky.
- G. K. Batchelor in 1946 wrote on Kolmogoroff’s theory of locally isotropic turbulence mentions tensors and vectors but does not give an elementary interpretation of Kolmogorov’s derivation. Can you?
At the end of the day, and quoting Bill George, the main issue with turbulence is that we lack a mathematical tool which can give us both accurate physical descriptions of flows but also capable of predicting properties of that flow. Any book on turbulence will inevitabley describe the Richardson-Kolmogorov turbulence cascade, which is arguably the most elementary interpretation of K41.
K41c is only three pages long and pretty straightforward: start with an equation for B_dd (eq. 3), the energy of eddies of size r or less, which involves a destruction term (E: turbulent dissipation) an term associated with energy exchange between scales (B_ddd: the third order structure function) and viscous diffusion (laplacian term). Massage this to get eq. 4 and then multiply by r^4 followed by integration from r=0 to r=R (note that at this point we assume E to be independent of r - see K62 if otherwise) so that you get eq. 5. For very small r, such that B_dd and B_ddd are Taylor exapandable, you find that for r->0 B_ddd must vanish faster than the remaining terms (it vanishes as r^3 whereas the other terms vanish as r), leading to eq. (6). For intermediate r (in the inertial range) let viscosity vanish to zero and obtain the 4/5 law (eq. (7) - not Richardson's 4/5 scaling).
If you're referring to "Double Velocity Correlation Function in Turbulent Motion" I'd argue the 3rd and 4th paragraphs have a pretty good (and elementary) interpretation of Kolmogorov's results - how often do you read a paper on turbulence without a single equation?
The following text is from the Turbulence Lecture notes of Prof. Ian Jacobi at Technion - Israel.
"Let’s consider a cloud of particles in a turbulent flow, e.g. smokes particles in the atmosphere. Let’s track two particles in this cloud. How fast do the particles separate from each other relative to the rate of motion of their joint center of mass, i.e. neglecting the net advection of the pair, or working from a moving reference frame fixed on their joint center of mass. If the particles spread by Fickian diffusion, then we can describe their spread by a diffusivity constant, D. We know that the squared separation distance between the particles, r^2, under the effect of Fickian diffusion alone, is then given by r^2 ∼ Dt This is referred to as normal diffusion and is a very slow process — the diffusivity of carbon dioxide in air is D ≈ 16 × 10-6 m^2/s so you can estimate the velocity dr/ dt . What happens when the mixing process is turbulent mixing instead of molecular diffusion? Let’s estimate the spreading rate in this case. Based on the argument above, the energy exchange between large and small scales in turbulence can be described entirely by the dissipation rate, \epsilon. The molecular effects, like viscosity did not matter. So we assume here too that molecular diffusivity doesn’t matter and that we can write the spreading of particules in our turbulent cloud as a function of only r^2(\epsilon, t) By dimensional analysis, we see that r^2 ∼ \epsilon t^3 This is therefore classified as ‘super-diffusion’ since the exponent of time is greater than one. We can also translate this ‘super-diffusion’ in terms of an effective diffusivity for turbulence, Dturb, by taking the time derivative of the mean squared separation dr^2/dt ∼ \epsilon t^2 ∼ \epsilon^1/3 r^4/3 So we find that instead of a constant diffusivity like in the case of Fick’s law, we have a scale-dependent effective diffusivity given by Dturb ∼ \epsilon^1/3 r^4/3 which is known as Richardson’s 4/3’s Law. "
I hope this will answer your question.
I don't think you'll find Richardson's 4/3 law explicitely derived in K41, however, you can derive it directly using some of the results from there: since your diffusivity depends on the velocity scale U associated with an eddy of size r, the velocity scale U can be obtained from B_dd (the second order structure function) or E_dd (the power spectral density), in any case, U is proportional to r^1/3 in the inertial range since B_dd~r^2/3 (or, equivalently E_dd~k^-5/3). You could do the same starting from C_\epsilon=\epsilon r / U^3=const and working your way back to a diffusivity coefficient.
Assuming you're talking about the 4/3 coefficients in eq. (24) in K41a and eq. (21) in K41c, these involve second order moments of the velocity structure function (not first and second) and this specific 4/3 appears purely a consequence of isotropy.
- What is the function B?
B is defined as a structure function, which in the context of K41 is just another measure for two-point correlations. Broadly speaking B is a real-space analog to the energy spectrum E. If E(k) characterises the energy associated with wavenumber k then B_dd(r) characterises the energy associated with a length scale r.
Take a homogenous, isotropic and statistically stationary flow and consider two points (say A and B) along a given axis (say x_d). If u is the velocity component in the direction x_d, then for every time instant/independent sample, you can compute (u_A - u_B)^p for some integer p, where u_A is the fluctuating velocity component at point A and u_B is the fluctuating velocity component at point B.
If you set p=2 and take the time/ensemble average () you obtain the second order structure function B_dd=, for p=3 you get the third order structure function B_ddd= and so on. The relevance of these quantities comes in the answer to your question below, but notice how these functions relate to the two-point correlation function: for example B_dd=2 -2 (because of homogeneity = = ); the analogy between the structure function and the correlation function means you can jump to E_dd by Fourier transforming B_dd/2- (see Wiener-Khinchin theorem).
Now consider v, the velocity component normal to x_d. We can define, as above, the structure functions B_nn=, B_nnn=, etc. B_nn and B_dd are related through isotropy (see e.g. https://arc.aiaa.org/doi/abs/10.2514/8.350).
Because K41 considers only homogenous, isotropic and statistically stationary flows, it suffices to consider a single velocity component in a single direction to fully describe the turbulence. For an assumption-free approach see https://arxiv.org/pdf/physics/0102063.pdf for structure functions and https://link.springer.com/article/10.1007/BF01329645 for energy spectra.
- What is the physical description corresponding to his mathematics?
In a crude way (see refs below), the velocity difference (u_A - u_B) is a characteristic velocity for eddies of size r_AB, with r_AB being the distance between points A and B. Thus, B_dd characterises the kinetic energy of eddies of size r_AB or smaller and B_ddd characterises the transport in the eddy-reference-frame of that kinetic energy.
Notice that these are analogous, but very different, to standard one-point measures for kinetic energy () and turbulent transport () as they include information about two-points (i.e. length scales) in the flow.
You can find extensive (and very good) discussions on the physical interpretation of these functions in turbulence text books such as Frisch's "Turbulence: The Legacy of A. N. Kolmogorov" and Davidson's "Turbulence: An Introduction for Scientists and Engineers". Or have a look at both Article Identifying Turbulent Energy Distributions in Real, Rather t...
and https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/on-the-deficiency-of-evenorder-structure-functions-as-inertialrange-diagnostics/20C72BC567F69524E1F25A1E6AEA19A5 which are very relevant to this question.One advantage of using structure functions as opposed to energy spectra is that they are somewhat more tenable/easy to define when dealing with inhomogeneous/anisotropic flows: for example, defining the spectrum near a solid boundary can be tricky.
- G. K. Batchelor in 1946 wrote on Kolmogoroff’s theory of locally isotropic turbulence mentions tensors and vectors but does not give an elementary interpretation of Kolmogorov’s derivation. Can you?
At the end of the day, and quoting Bill George, the main issue with turbulence is that we lack a mathematical tool which can give us both accurate physical descriptions of flows but also capable of predicting properties of that flow. Any book on turbulence will inevitabley describe the Richardson-Kolmogorov turbulence cascade, which is arguably the most elementary interpretation of K41.
K41c is only three pages long and pretty straightforward: start with an equation for B_dd (eq. 3), the energy of eddies of size r or less, which involves a destruction term (E: turbulent dissipation) an term associated with energy exchange between scales (B_ddd: the third order structure function) and viscous diffusion (laplacian term). Massage this to get eq. 4 and then multiply by r^4 followed by integration from r=0 to r=R (note that at this point we assume E to be independent of r - see K62 if otherwise) so that you get eq. 5. For very small r, such that B_dd and B_ddd are Taylor exapandable, you find that for r->0 B_ddd must vanish faster than the remaining terms (it vanishes as r^3 whereas the other terms vanish as r), leading to eq. (6). For intermediate r (in the inertial range) let viscosity vanish to zero and obtain the 4/5 law (eq. (7) - not Richardson's 4/5 scaling).
If you're referring to "Double Velocity Correlation Function in Turbulent Motion" I'd argue the 3rd and 4th paragraphs have a pretty good (and elementary) interpretation of Kolmogorov's results - how often do you read a paper on turbulence without a single equation?
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