Alternative formulations for statistical theories: 1.

In the spectral representation of turbulence it is well known that interactions in wavenumber space involve triads of wave vectors, with the members of each triad combining to form a triangle. It is perhaps less well known that the way in which this constraint is handled can have practical consequences. This was brought home to me in 1984, when we published our first calculations of the Local Energy Transfer (LET) theory [1].

Our goal was to compare the LET predictions of freely decaying isotropic turbulence with those of Kraichnan’s DIA, as first reported in 1964 [2]. With this in mind, we set out to calculate both DIA and LET under identical conditions; and also to compare out calculations of DIA with those of Kraichnan, in order to provide a benchmark. We applied the Edwards formulation [3] of the equations to both theories; but, apart from that, in order to ensure strict comparability we used exactly the same numerical methods as Kraichnan. Also, three of our initial spectral forms were the same as his, although we also introduced a fourth form to meet the suggestions of experimentalists when comparing with experimental results.

Reference should be made to [1] for details, but predictions of both theories were in line with experimental and numerical results in the field, with LET tending to give greater rates of energy transfer (and higher values of the evolved skewness factor) than DIA, which was assumed to be connected with its compatibility with the Kolmogorov spectrum. However, our calculation of the DIA value of the skewness was about 4% larger than Herring and Kraichnan found [4], which could only be explained by the different mathematical formulation.

Let us consider the two different ways of handling the wavenumber constraint, as follows.

Kraichnan’s notation involved the three wave vectors $\mathbf{k}$, $\mathbf{p}$, and $\mathbf{q}$; and used the identity: \begin{equation}\int d^3p\int d^3q \,\delta(\mathbf{k}-\mathbf{p}-\mathbf{q})f(k,p,q)=\int_{p+q=k}dpdq\frac{2\pi pq}{k}f(k,p,q), \end{equation} where the constraint is expressed by the Dirac delta function and $ f(k,p,q)$ is some relevant function. Note that the domain of integration is in the $(p,q)$ plane, such that the condition $p+q=k$ is always satisfied.

Edwards [3] used a more conventional notation of $\mathbf{k}$, $\mathbf{j}$, and $\mathbf{l}$; and followed a more conventional route of simply integrating over one of the dummy wave vectors in order to eliminate the delta function, thus: \begin{equation}\int d^3j\int d^3l \,\delta(\mathbf{k}-\mathbf{j}-\mathbf{l})f(k,j,l)=\int_0^\infty 2\pi j^2 dj\int_{-1}^{1}d\mu f(k,j,|\mathbf{k}-\mathbf{j}|),\end{equation}where $\mu = \cos \theta_{kj}$ and $\theta_{kj}$ is the angle between the vectors $\mathbf{k}$ and $\mathbf{j}$.

Of course the two formulations are mathematically equivalent. Where differences arise is in the way they handle rounding and truncation errors in numerical procedures. It was pointed out by Kraichnan [2], that corrections had to be made when triangles took the extreme form of having one side very much smaller than the other two. If this problem can lead to an error of about 4%, then it is worth investigating further. I will enlarge on this matter in my next post.

[1] W. D. McComb and V. Shanmugasundaram. Numerical calculations of decaying isotropic turbulence using the LET theory. J. Fluid Mech., 143:95-123, 1984.
[2] R. H. Kraichnan. Decay of isotropic turbulence in the Direct-Interaction Approximation. Phys. Fluids, 7(7):1030{1048, 1964.
[3] S. F. Edwards. The statistical dynamics of homogeneous turbulence. J. Fluid Mech., 18:239, 1964.
[4] J. R. Herring and R. H. Kraichnan. Comparison of some approximations for isotropic turbulence Lecture Notes in Physics, volume 12, chapter Statistical Models and Turbulence, page 148. Springer, Berlin, 1972.

From $k^{-5/3}$ to $x^{2/3}$.

From time to time, I have remarked that all the controversy about Kolmogorov’s (1941) theory arises because his real-space derivation is rather imprecise. A rigorous derivation relies on a wavenumber-space treatment; and then, in principle, one could derive the two-thirds law for the second-order structure function from Fourier transformation of the minus five-thirds law for the energy spectrum. However, the fractional powers can seem rather daunting and when I was starting out I was fortunate to find a neat way of dealing with this in the book by Hinze [1].

We will work with $E_1(k_1)$, the energy spectrum of longitudinal velocity fluctuations, and $f(x_1)$, the longitudinal correlation coefficient. Hinze [1] cites Taylor [2] as the source of the cosine-Fourier transform relationship between these two quantities, thus:\begin{equation}U^2 f(x) = \int_0^\infty\, dk_1\,E_1(k_1) \cos(k_1x_1),\end{equation}and \begin{equation}E_1(k_1) =\frac{2}{\pi}\int_0^\infty\, dx_1\, f(x_1) \cos(k_1x_1),\end{equation} where $U$ is the root mean square velocity.

In general, the power laws only apply in the inertial range, which means that we need to restrict the limits of the integrations. However, Hinze obtained a form which allows one to work with the definite limits given above, and reference should be made to page 198 of the first edition of his book [1] for the expression: \begin{equation}U^2\left[1-f(x_1)\right] = C \int_0^\infty\,dk_1\,k_1^{-5/3}\left[1-\cos(k_1x_1)\right],\label{hinze} \end{equation} where $C$ is a universal constant.

The trick he employed to evaluate the right hand side is to make the change of variables:\begin{equation}y= k_1x_1 \quad \mbox{hence} \quad dk_1 =\frac{dy}{x_1}. \end{equation} With this substitution, the right hand side of equation (\ref{hinze}) becomes: \begin{equation}\mbox{RHS of (3)} = C x_1^{2/3}\,\int_0^\infty \,dy [1-\cos y].\end{equation} Integration by parts then leads to: \begin{equation}\int_0^\infty \,dy [1-\cos y]=\frac{3}{2}\int_0^\infty\, dy\, y^{-2/3}\, \sin y =\frac{3}{4}\Gamma(1/3),\end{equation} where $\Gamma$ is the gamma function. Note that I have omitted any time dependence for sake of simplicity, but of course this is easily added.

[1] J. O. Hinze. Turbulence. McGraw-Hill, New York, 1st edition, 1959. (2nd edition, 1975).
[2] G. I. Taylor. Statistical theory of turbulence. Proc. R. Soc., London, Ser.A, 151:421, 1935.

Compatibility of temporal spectra with Kolmogorov (1941) and with random sweeping.

I previously wrote about temporal frequency spectra, in the context of the Taylor hypothesis and a uniform convection velocity of $U_c$, in my post of 25 February 2021. At the time, I said that I would return to the more difficult question of what happens when there is no uniform convection velcocity present. I also said that this would not necessarily be next week, so at least I was right about that.

As in the earlier post, we consider a turbulent velocity field $u(x,t)$ which is stationary and homogeneous with rms value $U$. This time we just consider the dimensions of the temporal frequency spectrum $E(\omega)$. We use the angular frequency $\omega = 2\pi n$, where $n$ is the frequency in Hertz, in order to be consistent with the usual definition of wavenumber $k$. Integrating the spectrum, we have the condition: \begin{equation}\int_0^\infty E(\omega) d\omega = U^2,\end{equation}which gives us the dimensions: \begin{equation}\mbox{Dimensions of}\; E(\omega)d\omega = L^2 T^{-2};\end{equation} or velocity squared.

For many years, the literature relating to the wavenumber-frequency correlation $C(k,\omega)$ has been dominated by the question: is decorrelation due to random sweeping effects, which would mean that the characteristic time is the sweeping timescale $(Uk)^{-1}$; or is it characterised by the Kolmogorov timescale $(\varepsilon^{1/3}k^{2/3})^{-1}$? A recent article [1] makes a typical point about the consequences for the frequency spectrum of the dominance of the sweeping effect: ‘… the frequency energy spectrum of Eulerian velocities exhibits a $\omega^{-5/3}$ decay, instead of the $\omega^{-2}$ expected from K41 scaling’. Which is counter-intuitive at first sight! As we saw in my blog of 26/02/21, for the case of uniform convection $\omega^{-5/3}$ is associated with K41.

Let us begin by clearing up the latter point. The authors of [1] cite the book by Monin and Yaglom, but I was unable to find it. (I mean the reference, not the book which is quite conspicuous on my bookshelves. I think that anyone giving a reference to a book, should cite the page number. Sometimes I do that and sometimes I forget!) In any case, it is easy enough to work out. From equation (2) we have the dimensions of $E(\omega)$ as $L^{2}T^{-1}$. From the K41 approach we can write for the inertial range: \begin{equation}E(\omega) \sim \varepsilon^{n}\omega^{m} \sim \varepsilon \omega^{-2},\end{equation} where we fixed the dependence on the index $n$ first.

The interest in random convective sweeping mainly stems from Kraichnan’s analysis of his direct-interaction approximation (DIA), dating back to 1959. A general discussion of this will be found in the book [2], but we can take a shortcut by noting that Kraichnan obtained an approximate solution for the reponse function $G(k,\tau)$ of his theory (see page 219 of [2]) as: \begin{equation}G(k,\tau)=\frac{exp(-\nu k^2\tau)J_1(2Uk\tau)}{Uk\tau},\end{equation} where $\tau = t-t’$, $\nu$ is the kinematic viscosity, and $J_1$ is a Bessel function of the first kind. The interesting thing about this is that the K41 characteristic time for the inertial range does not appear. Also, in the inertial range, the exponential factor can be put to one, and the decay is determined by the sweeping time $(Uk)^{-1}$.

Corresponding to this solution for the inertial range, the energy spectrum takes the form: \begin{equation} E(k) \sim (\varepsilon U)^{1/2}k^{-3/2},\end{equation} as given by equation (6.50) in [2]. As is well known, this $-3/2$ law is sufficiently different from the observed form, which is generally compatible with the K41 $-5/3$ wavenumber spectrum, to be regarded as incorrect. We can obtain the frequency spectrum corresponding to the random sweeping hypothesis by simply replacing the convective velocity $U_c$, as used in Taylor’s hypothesis, by the rms velocity $U$. From equation (8) of the earlier blog, we have; \begin{equation}E(\omega) \sim (\varepsilon U_c)^{2/3}\omega^{-5/3} \rightarrow (\varepsilon U)^{2/3}\omega^{-5/3} , \quad \mbox{when} \quad U_c \rightarrow U. \end{equation}
This result is rather paradoxical to say the least. In order to get a $-5/3$ dependence on frequency, we have to have a $-3/2$ dependence on wavenumber! It is many years since I looked into this and in view of the continuing interest in the subject, I have begun to rexamine it. For the moment, I would make just one observation.

Invoking Taylor’s expression for the dissipation rate, which is: $\varepsilon = C_\varepsilon U^3/L$, where $L$ is the integral lengthscale (not to be confused with the symbol for the length dimension) and $C_\varepsilon$ asymptotes to a constant value for Taylor-Reynolds numbers $R_\lambda \sim 100$ [3], we may examine the relationship between the random sweeping and K41 timescales. Substituting for the rms velocity, have: \begin{equation}\tau_{sweep} =(Uk)^{-1}\sim (\varepsilon^{1/3}L^{1/3}k)^{-1}.\end{equation} Then, putting $k\sim 1/L \equiv k_L$, we obtain:\begin{equation}\tau_{sweep}\sim (\varepsilon^{1/3}k_L^{2/3})^{-1} = \tau_{K41}(k_L).\end{equation} So the random sweeping timescale becomes equal to the K41 timescale for wavenumbers in the energy-containing range. Just to make things more puzzling!

[1] A. Gorbunova, G. Balarac, L. Canet, G. Eyink, and V. Rossetto. Spatiotemporal correlations in three-dimensional homogeneous and isotropic turbulence. Phys. Fluids, 33:045114, 2021.
[2] W. D. McComb. The Physics of Fluid Turbulence. Oxford University Press, 1990.
[3] W. D. McComb, A. Berera, S. R. Yoffe, and M. F. Linkmann. Energy transfer and dissipation in forced isotropic turbulence. Phys. Rev. E, 91:043013, 2015.