Summary of the Kolmogorov-Obukhov (1941) theory. Part 3: Obukhov’s theory in k-space.

Summary of the Kolmogorov-Obukhov (1941) theory. Part 3: Obukhov’s theory in k-space.

Obukhov is regarded as having begun the treatment of the problem in wavenumber space. In [1] he referred to an earlier paper by Kolmogorov for the spectral decomposition of the velocity field in one dimension and pointed out that the three-dimensional case is carried out similarly by multiple Fourier integrals. He employed the Fourier-Stieltjes integral but fortunately this usage did not survive. For many decades the standard Fourier transform has been employed in this field.

[a] Obukhov’s paper [1] was published between K41A and K41B, and was described by Batchelor ‘as to some extent anticipating the work of Kolmogorov’. He worked with the energy balance in $k$-space and, influenced by Prandtl’s work, introduced an ad hoc closure based on an effective viscosity.

[b] The derivation of the ‘-5/3’ law for the energy spectrum seems to have been due to Onsager [2]. He argued that Kolmogorov’s similarity principles in $x$-space would imply an invariant flux (equal to the dissipation) through those wavenumbers where the viscosity could be neglected. Dimensional analysis then led to $E(k) \sim \varepsilon^{2/3}k^{-5/3}$.

[c] As mentioned in the previous post (points [c] and [d]), Batchelor discussed both K41A and K41B in his paper [3], but did not include K41B in his book [4]. Also, in his book [4], he discussed K41A entirely in wavenumber space. The reasons for this change to a somewhat revisionist approach can only be guessed at, but there may be a clue in his book. On page 29, first paragraph, he says: ‘Fourier analysis of the velocity field provides us with an extremely valuable analytical tool and one that is well-nigh indispensable for the interpretation of equilibrium or similarity hypotheses.’ (The emphasis is mine.)

[d] This is a very strong statement, and of course the reference is to Kolmogorov’s theory. There is also the fact that K41B is not easily translated into $k$-space. Others followed suit, and Hinze [5] actually gave the impression of quoting from K41A but used the word ‘wavenumber’, which does not in fact occur in that work. By the time I began work as a postgraduate student in 1966, the use of spectral methods had become universal in both experiment and theory.

[e] There does not appear to be any $k$-space ad hoc closure of the Lin equation to parallel K41B (i.e. the derivation of the ‘4/5’ law); but, for the specific case of stationary turbulence, I have put forward a treatment which uses the infinite Reynolds number limit to eliminate the energy spectrum, while retaining its effect through the dissipation rate [6]. It is based on the scale invariance of the inertial flux, thus: \begin{equation}\Pi(\kappa)=-\int_0^{\kappa}dk\,T(k) = \varepsilon, \end{equation}which of course can be written in terms of the triple-moment of the velocity field. As the velocity field in $k$-space is complex, we can write it in terms of amplitude and phase. Accordingly, \begin{equation}u_{\alpha}(\mathbf{k}) = V(\kappa)\psi_{\alpha}(k’),\end{equation} where $V(\kappa)$ is the root-mean-square velocity, $k’=k/\kappa$ and $\psi$ represents phase effects. The result is: \begin{equation}V(\kappa)=B^{-1/3}\varepsilon^{1/3}\kappa^{-10/3},\end{equation}where $B$ is a constant determined by an integral over the triple-moment of the phases of the system. The Kolmogorov spectral constant is then found to be: $4\pi\, B^{-2/3}$.

[f] Of course a statistical closure, such as the LET theory, is needed to evaluate the expression for $B$. Nevertheless, it is of interest to note that this theory provides an answer to Kraichnan’s interpretation of Landau’s criticism of K41A [7]. Namely, that the dependence of an average (i.e. the spectrum) on the two-thirds power of an average (i.e. the term involving the dissipation) destroys the linearity of the averaging process. In fact, the minus two-thirds power of the average in the form of $B^{-2/3}$ cancels the dependence associated with the dissipation.

[1] A. M. Obukhov. On the distribution of energy in the spectrum of turbulent flow. C.R. Acad. Sci. U.R.S.S, 32:19, 1941.
[2] L. Onsager. The Distribution of Energy in Turbulence. Phys. Rev., 68:281, 1945. (Abstract only.)
[3] G. K. Batchelor. Kolmogoroff’s theory of locally isotropic turbulence. Proc. Camb. Philos. Soc., 43:533, 1947.
[4] G. K. Batchelor. The theory of homogeneous turbulence. Cambridge University Press, Cambridge, 1st edition, 1953.
[5] J. O. Hinze. Turbulence. McGraw-Hill, New York, 2nd edition, 1975. (First edition in 1959.)
[6] David McComb. Scale-invariance and the inertial-range spectrum in three-dimensional stationary, isotropic turbulence. J. Phys. A: Math. Theor., 42:125501, 2009.
[7] R. H. Kraichnan. On Kolmogorov’s inertial-range theories. J. Fluid Mech., 62:305, 1974.