Fashions in turbulence theory.

Back in the 1980s, fractals were all the rage. They were going to solve everything, and turbulence was no exception. The only thing that I can remember from their use in microscopic physics was that the idea was applied to the problem of diffusion-limited aggregation, and I’ve no memory of how successful they were (or were not). In turbulence they were a hot topic for solving the supposed problem of intermittency, and there was a rash of papers decorated with esoteric mathematical terms. This could be regarded as ‘Merely corroborative detail, intended to give artistic verisimilitude to an otherwise bald and unconvincing narrative.’[1]. When these proved inadequate, the next step was multifractals, which rather underlined the fact that this approach was at best a phenomenology, rather than a fundamental theory. And that activity too seems to have died away.

Another fashion of the 1970s/80s was the idea of deterministic chaos. This began around 1963 with the Lorentz system, a set of simple differential equations intended to model atmospheric convection. These equations were readily computed, and it was established that their solutions were sensitive to small changes in initial conditions. With the growing availability of desktop computers in the following decades, low-dimensional dynamical systems of this kind provided a popular playground for mathematicians and we all began to hear about Lorentz attractors, strange attractors, and the butterfly effect. Just to make contact with the previous fashion, the phase space portraits of these systems often were found to have a fractal structure!

In 1990, a reviewer of my first book [2] rebuked me for saying so little about chaos and asserted that it would be a dominant feature of turbulence theory in the future. Well, thirty years on and we are still waiting for it. The problem with this prediction is that turbulence, in contrast to the low-dimensional models studied by the chaos enthusiasts, involves large numbers of degrees of freedom; and these are all coupled together. As a consequence, the average behaviour of a turbulent fluid is really quite insensitive to fine details of its initial conditions. In reality that butterfly can flap its wings as much as it likes, but it isn’t going to cause a storm.

In fairness, although we have gone back to using our older language of ‘random’ rather than ‘chaotic’ when studying turbulence, the fact remains that deterministic chaos is actually a very useful concept. This is particularly so when taken in the context of complexity, and that will be the subject of our next post.

[1] W. S. Gilbert, The Mikado, Act 2, 1852.
[2] W. D. McComb. The Physics of Fluid Turbulence. Oxford University Press, 1990.

Summary of the Kolmogorov-Obukhov (1941) theory: overview.

In the last three posts we have summarised various aspects of the Kolmogorov-Obukhov (1941) theory. When considering this theory, the following things need to be borne in mind.

[a] Whether we are working in $x$-space or $k$-space matters. See my posts of 8 April and 15 April 2021 for a concise general discussion.
[b] In $x$-space the equation of motion (NSE) simply presents us with the problem of an insoluble, nonlinear partial differential equation.
[c] In $k$-space the NSE presents a problem in statistical physics and in itself tells us much about the transfer and dissipation of turbulent kinetic energy.
[d] The Karman-Howarth equation is a local energy balance that holds for any particular value of the distance $r$ between two measuring points.
[e] There is no energy flux between different values of $r$; or, alternatively, through scale.
[f] The energy flux $\Pi(k)$ is derived from the Lin equation (i.e. in wavenumber space) and cannot be applied in $x$-space.
[g] The maximum value of the energy flux, $\Pi_{max}=\varepsilon_T$ (say), is a number, not a function, and can be used (like the dissipation $\varepsilon$) in both $k$-space and $x$-space.
[h] It also matters whether the isotropic turbulence we are considering is stationary or decaying in time.
[g] If the turbulence is decaying in time, then K41B relies on Kolmogorov’s hypothesis of local stationarity. It has been pointed out in a previous post (Part 2 of the present series) that this cannot be the case by virtue of restriction to a range of scales nor in the limit of infinite Reynolds number [1]. See also the supplemental material for [2].
[h] In $k$-space this is not a problem and the $k^{-5/3}$ spectrum can still be expected [1], as of course is found in practice.
[i] If the turbulence is stationary, then K41B is exact for a range of wavenumbers for sufficiently large Reynolds numbers. The extent of this inertial range increases with increasing Reynolds numbers.

I have not said anything in this series about the concept of intermittency corrections or anomalous exponents. This topic has been dealt with in various blogs and soon will be again.

[1] W. D. McComb and R. B. Fairhurst. The dimensionless dissipation rate and the Kolmogorov (1941) hypothesis of local stationarity in freely decaying isotropic turbulence. J. Math. Phys., 59:073103, 2018.
[2] W. David McComb, Arjun Berera, Matthew Salewski, and Sam R. Yoffe. Taylor’s (1935) dissipation surrogate reinterpreted. Phys. Fluids, 22:61704, 2010.

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.

Summary of the Kolmogorov-Obukhov (1941) theory. Part 2: Kolmogorov’s theory in x-space.
Kolmogorov worked in $x$-space and his two relevant papers are cited below as [1] (often referred to as K41A) and [2] (K41B). We may make a pointwise summary of this work, along with more recent developments as follows.

[a] In K41A, Kolmogorov introduced the concepts of local homogeneity and local isotropy, as applying in a restricted range of scales and in a restricted volume of space. He also seems to have introduced what we now call the structure functions, allowing the introduction of scale through the correlations of velocity differences taken between two points separated by a distance $r$. He used Richardson’s concept of a cascade of eddies, in an intuitive way, to introduce the idea of an inertial sub-range, and then used dimensional analysis to deduce that (in modern notation) $S_2 \sim \varepsilon^{2/3}r^{2/3}$.

[b] In K41B, he used an ad hoc closure of the Karman-Howarth equation (KHE) to argue that $S_3 = 0.8 \varepsilon r$ in the inertial range of values of $r$: the well-known ‘four-fifths law’. He further assumed that the skewness factor was constant and found that this led to the K41A result for $S_2$. The closure was based on the fact that the term explicit in $S_2$ would vanish as the viscosity tended to zero, whereas its effect could still be retained in the dissipation rate.

[c] In 1947, Batchelor [3] provided an exegesis of both these theories. In the case of K41A, this was only partial, but he did make it clear that K41A relied (at least implicitly) on Richardson’s idea of the ‘eddy cascade’. He also pointed out that K41B could not be readily extended to higher-order equations in the statistical hierarchy, because of the presence of the pressure term with its long-range properties in the higher-order equations.

[d] Moffatt [4] credited this paper by Batchelor with bringing Kolmogorov’s work to the Western world. He also, in effect, expressed surprise that Batchelor did not include his re-derivation of K41B in his book. This is a very interesting point; and, in my view, is not unconnected to the fact that Batchelor discussed K41A almost entirely in wavenumber space in his book. I will return to this later.

[e] In 2002, Lundgren re-derived the K41B result, by expanding the dimensionless structure functions in powers of the inverse Reynolds number. By demanding that the expansions matched asymptotically in an overlap region between outer and inner scaling regimes, he was also able to recover the K41A result without the need to make an additional assumption about the constancy of the skewness.

[f] More recently, McComb and Fairhurst [6] used the asymptotic expansion of the dimensionless structure functions to test Kolmogorov’s hypothesis of local stationarity and concluded that it could not be true. They found that the time-derivative must give rise to a constant term; which, however small, violates the K41B derivation of the four-fifths law. Nevertheless, they noted that in wavenumber space, this term (which plays the part of an input to the KHE) will appear as a Dirac delta function at the origin, and hence does not violate the derivation of the minus five-thirds law in $k$-space. We will extend this idea further in the next post.

[1] A. N. Kolmogorov. The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. C. R. Acad. Sci. URSS, 30:301, 1941. (K41A)
[2] A. N. Kolmogorov. Dissipation of energy in locally isotropic turbulence. C. R. Acad. Sci. URSS, 32:16, 1941. (K41B)
[3] G. K. Batchelor. Kolmogoroff’s theory of locally isotropic turbulence. Proc. Camb. Philos. Soc., 43:533, 1947.
[4] H. K. Moffatt. G. K. Batchelor and the Homogenization of Turbulence. Annu. Rev. Fluid Mech., 34:19-35, 2002.
[5] Thomas S. Lundgren. Kolmogorov two-thirds law by matched asymptotic expansion. Phys. Fluids, 14:638, 2002.
[6] W. D. McComb and R. B. Fairhurst. The dimensionless dissipation rate and the Kolmogorov (1941) hypothesis of local stationarity in freely decaying isotropic turbulence. J. Math. Phys., 59:073103, 2018.

Summary of Kolmogorov-Obukhov (1941) theory. Part 1: some preliminaries in $x$-space and $k$-space.

Discussions of the Kolmogorov-Obukhov theory often touch on the question: can the two-thirds law; or, alternatively, the minus five-thirds law, be derived from the equations of motion (NSE)? And the answer is almost always: ‘no, they can’t’! Yet virtually every aspect of this theory is based on what can be readily deduced from the NSE, and indeed has so been deduced, many years ago. So our preliminary here to the actual summary, is to consider what we know from a consideration of the NSE, in both $x$-space and $k$-space. As another preliminary, all the notation is standard and can be found in the two books cited below as references.

We begin with the familiar NSE, consisting of the equation of motion, \begin{equation}\frac{\partial u_{\alpha}}{\partial t} + \frac{\partial (u_{\alpha}u_\beta)}{\partial x_\beta} =-\frac{1}{\rho}\frac{\partial p}{\partial x_\alpha} + \nu \nabla^2 u_\alpha,\end{equation}which expresses conservation of momentum and is local, in that it gives the relationship between the various terms at one point in space; and the incompressibility condition \begin{equation}\frac{\partial u_\beta}{\partial x_\beta} = 0.\end{equation} It is well known that taking these two equations together allows us to eliminate the pressure by solving a Poisson-type equation. The result is an expression for the pressure which is an integral over the entire velocity field: see equations (2.3) and (2.9) in [1].

In $k$-space we may write the Fourier-transformed version of (1) as: \begin{equation}\frac{\partial u_\alpha(\mathbf{k},t)}{\partial t} + i k_\beta\int d^3 j u_\alpha(\mathbf{k-j}.t)u_\beta(\mathbf{j},t) = k_\alpha p(\mathbf{k},t) -\nu k^2 u_\alpha (\mathbf{k},t).\end{equation} The derivation can be found in Section 2.4 of [2]. Also, the discrete Fourier-series version (i.e. in finite box) is equation (2.37) in [2].

The crucial point here is that the modes $\mathbf{u}(\mathbf{k},t)$ form a complete set of degrees of freedom and that each mode is coupled to every other mode by the non-linear term. So this is not just a problem in statistical physics, it is an example of the many-body problem.

Note that (1) gives no hint of the cascade, but (3) does. All modes are coupled together and, if there were no viscosity present, this would lead to equipartition, as the conservative non-linear term merely shares out energy among the modes. The viscous term is symmetry-breaking due to the factor $k^2$ which increases the dissipation as the wavenumber increases. This prevents equipartition and leads to a cascade from low to high wavenumbers. All of this becomes even clearer when we multiply the equation of motion by the velocity and average. We then obtain the energy-balance equations in both $x$-space and $k$-space.

We begin in real space with the Karman-Howarth equation (KHE). This can be written in various forms (see Section 3.10.1 in [2]), and here we write in terms of the structure functions for the case of free decay: \begin{equation}\varepsilon =-\frac{3}{4}\frac{\partial S_2}{\partial t}+\frac{1}{4r^4}\frac{\partial (r^4 S_3)}{\partial r} +\frac{3\nu}{2r^4}\frac{\partial}{\partial r}\left(r^4\frac{\partial S_2}{\partial r}\right).\end{equation}Note that the pressure does not appear, as a correlation of the form $\langle up \rangle$ cannot contribute to an isotropic field, and that strictly the left hand side should be the decay rate $\varepsilon_D$ but it is usual to replace this by the dissipation as the two are equal in free decay. Full details of the derivation can be found in Section 3.10 of [2].

For our present purposes, we should emphasise two points. First, this is one equation for two dependent variables and so requires a statistical closure in order to solve for one of the two. In other words, it is an instance of the notorious statistical closure problem. Second, it is local in the variable $r$ and does not couple different scales together. It holds for any value of $r$ but is an energy balance locally at any chosen value of $r$.

The Lin equation is the Fourier transform of the KHE. It can be derived directly in $k$-space from the NSE (see Section 3.2.1 in [2]):\begin{equation} \left(\frac{\partial}{\partial t} + 2\nu k^2\right)E(k,t) = T(k,t).\end{equation} Here $T(k,t)$ is called the transfer spectrum, and can be written as: \begin{equation}T(k,t)= \int_0^\infty dj\, S(k,j:t),\end{equation}where $S(k,j;t)$ is the transfer spectral density and can be expressed in terms of the third-order moment $C_{\alpha\beta\gamma}(\mathbf{j},\mathbf{k-j},\mathbf{-k};t)$.

Unlike the KHE, which is purely local in its independent variable, the Lin equation is non-local in wavenumber. We can define its associated inter-mode energy flux as:\begin{equation}\Pi (\kappa,t) = \int_\kappa^\infty dk\,T(k,t) = -\int_0^\kappa dk \, T(k,t).\end{equation}

We have now laid a basis for a summary of the Kolmogorov-Obhukov theory and one point should have emerged clearly: the energy cascade is well defined in wavenumber space. It is not defined at all in the context of energy conservation in real space. It can only exist as an intuitive phenomenon which is extended in space and time.

[1] W. D. McComb. The Physics of Fluid Turbulence. Oxford University Press, 1990.
[2] W. David McComb. Homogeneous, Isotropic Turbulence: Phenomenology, Renormalization and Statistical Closures. Oxford University Press, 2014.