Let us analyze in more details the results of the present theory that we shall call for reference NT, a new theory.
9.1 Fractal dimension and the number of independent ve- locity harmonics.
The Eq. (8.59) shows that the fractal dimension of the sub- domain with reduced nonlinear coupling is DF = 2.5. This is a
sub-domain that is actually turbulence itself because most probably all the fluid flow in 3D space is generated by the vorticity field in this sub-domain. Accordingly the number of Fourier flow harmonics to describe the flow is:
Nmoderen ≈(kd/k0)3(kd/k0)−µ = (kd/k0)5/2=Re15/8, (9.1)
where the definition (2.16) was used. As expected the real number of independent velocity harmonics is much less than in K41 theory given by (2.22): Nmoderen << Nmode. Although the above derivations
did not use explicitly the helical nature of the sub-domain with re- duced nonlinear coupling there is no other conceivable scenario but
BCCto explain it. It is rather the other way around so that it is the discovery of BCCthat allowed the bold assumptions in this section to be made for derivation of (8.59). Most of theNmodeturbulence ve-
locity harmonics in the ultraviolet range are locked up in the helicity fluctuations.
9.2 The K41 spectrum in the inertial range.
The reality is that apparently even in the inertial range the energy spectrum is not exactly K41. This is because of the non-universal log- arithmic correction (8.64). Even though we chose the K41 spectrum as a starting ”free solution” the RNG perturbation theory generated logarithmic corrections.62 Indeed, the corresponding energy spec- trum is now:
E(k)∝k−5/3(lnk/k0)−1/2. (9.2)
62Such logarithmic corrections are typical for RNG developed asymptotically
free ultraviolet solutions in quantum field theories. Here the solution is of course more complicated since the starting differential equations and the subsequent functional integral representations are more complicated
The logarithmic correction can be behind two facts that are quoted in Mininni et.al. (2008b). The first one is that even when the inertial range is identified in DNS the energy spectrum there has a slightly steeper slope than that for −5/3 power law. The second fact is the value of the constant in front of the K41 spectrum. It decreases no doubt with the increase of the Reynolds number of the simula- tions. Clearly from (9.2) it would follow that this ”constant” is in fact asymptotically Reynolds number dependent. But of course DNS over a wider range and much higher Reynolds numbers DNS should be carried out for definitive conclusions.
9.3 The K41 spectrum, inertial range and the buffer zone.
The origin of the K41 spectrum and the inertial range are also quite different from the K41 theory.63 Apart from the fact that the
K41 spectrum is formed by a different mechanism, actually by a small sub-domain of intensive vorticity in physical space, the inertial range ink-spacewhere it is likely to be true at least approximately, is effec- tively different. Although in the ultraviolet RNG theory developed above there is no upper cutoff it is clear that there must be a matching in k-space with the range where the viscous dissipation is dominant. In the K41 theory the inertial range was (2.21) and actually extended to kd. However NT indicates that that the inertial range is likely to
be shorter; to be sure for the high values of for which one expects the inertial range to exist at all. Most of the harmonics are locked up now inBCCand the number of independent harmonics is:
kd0 ∼kd(k0/kd)−µ/3∼kdRe−1/8. (9.3)
It should be added that the definition of kd itself is independent of
K41 theory. It just follows from the condition (3.25) that is an exact consequence of the Navier-Stokes equations and thus remains intact. But substituting the K41 spectrum into (3.25) and carry out the in- tegration up to k ≤ kd would be wrong in a way. Because in the
intervalkd0 < k < kdthe energy spectrum is not K41. But it does not
really matter if our attitude to (3.25) is as just defining certain value
63It is remainded that K41 spectrum does not necessarily mean the full K41
in the interval of wavenumbers ink-spacethat are much higher than the inertial range wavenumbers. In order to preserve the global value of enstrophy Ω =< ω2 >it is necessary that most of enstrophy and
energy dissipation is located in the part ofk-spacethat is more ultra- violet than the inertial range. What emerges is the energy spectrum structure that is more complicated than in the K41 theory. In the latter the inertial range smoothly merges with the viscous dissipation range for which the energy spectrum is exponentially decreasing with
k. In the present theory there should be an intermediate buffer zone connecting the inertial range with the K41 spectrum and the viscous sub-range k ≥kd, with the exponentially fast decreasing spectrum,
in which the viscosity effects and the nonlinear coupling are of the same order.
It should be noted that in the boundary layer (BL) turbulence the existence of an empirical buffer zone that separates the universal Prandtl logarithmic mean velocity profile and the viscous sub- range, all in physical space of course, is well known and fundamental. It is in the buffer zone where most of vortical CS are generated and most of energy is dissipated. There is a well known and far reaching relation between the logarithmic mean velocity profile in physical space for BL turbulence and the inertial range K41 spectrum in k- space (e.g., Levich, 1996). It is conjectured that the buffer zone in
k-space has much similar in nature with the buffer zone in physical space for BL turbulence. I shall return to BL turbulence, the one most importance in practical applications in the next section. In the meantime it is suggested that in HIT most of energy dissipation and enstrophy birth are located not in the K41 part of the spectrum but in a more ultraviolet part of k-space that I call the buffer zone by analogy with BL turbulence.
9.4 Analysis of the functional integral representation (8.13) in the inertial range.
Let us go back to the functional integral representation and look how the results (8.59)-(8.62) fit in. Consider again the generating functional (8.22) let us substitute the results (8.59)-(8.62) into it and try to estimate the contributions of various terms in the sense of mean field approximation, i.e., approximating the functional in the
exponent in (8.22) by the mean:
<
Z
dDkdfΦ−1{|X|2+<<|X|2>>}. (9.4)
In other words we shall assume that (8.59)-(8.62) are the most proba- ble flow realizations dominant in their contribution to the total func- tional (8.22). Since thesingular contribution is of the same order of magnitude as theregular one by virtue of (8.24) it is enough to esti- mate<R
dDdfΦ−1|X
reg|2>. There are three types of terms: bilinear
BL, nonlinear NL and proportional to viscosity V(k) The meaning of the mean field approximation now would be that we consider only the most contributing velocity field realizations that we assume to be the ones resulting in (8.59)-(8.62). The bilinear terms are:
BL(k) =
Z k
k0
dDdfΦ−1f2<v(k,f)|2>, (9.5)
where the integration is in a slice ofk-spacein the inertial range with
k→ ∞. The nonlinear terms are (with tensor indices and irrelevant for the scaling analysis constant factors omitted):
N L(k) = = Z k k0 dDdfΦ−1[λren]2(k)P(k)2<| Z k k0 dDdf ’v(k-q,f-f ’v(q,f ’|2> (9.6) And the viscous terms are:
V=ν2|intkk0k4Φ−1<|v(k,f)|2> dDdf. (9.7)
The cross terms in the integrand in the functional (8.22) are ne- glected. Their contribution can be easily shown to be of the same order as from the nonlinear terms that were picked up for the anal- ysis. Now we must estimate what are the scaling powers ofBLand
NL, i.e. to calculate [BL(k)] and [NL(k)]. The calculation should be done for two cases, the K41 theory and the New Theory (NT). Let us start fromBL. The powers counting yields:
[BL(k)] =
Z k
k0
=D+ [f] + [Φ−1] + 2[f] + [|v(k,f)|2] = 0, (9.8)
where the following relations were used: [Φ−1] =D−µ/3;
[f] = 2/3 +µ/3; (9.9) [<|v(k,f)|2>] = [E(k,f)] + [δ(k] +δ(f] = [E(k)]−[f]−2−D−[f] =
=−5/3−(D+ 2)−2[f].
It is noticed that as it should be, [BL(k)] = 0, for both K41 theory, i.e. forµ= 0 and for anyµ6= 0. The meaning of this analysis is that the contribution of turbulent realizations corresponding to the K41 spectrum is indeed satisfactory as far as theBLterms are concerned since with a logarithmic accuracy it does not depend on any of the cutoffs, eitherk0 or kd. But sinceBL correspond to the linear term
in the Navier-Stokes equations this result is trivial. TheNL(k) terms are the ones that are important.
The situation is different withNL(k). The calculation of [NL(k)] of course cannot be carried out exactly because it would be equivalent in a sense to the calculation of the functional integral itself. In reality this can be done only by RNG perturbation expansion that is quite equivalent to the perturbative solution of the model equations (8.30). But with certain assumptions an estimate can be done for NL(k) as it is defined in (9.6). Let us rewrite the (9.6) by doing the reverse Fourier transformation of the nonlinear part of Xi0(k,f) in (8.14) in the following equivalent manner:
N L(k) = Z k k0 Φ(k)−1dD Z <J(r,0)J(r0, t)> eik·(r-r’dDrdDr0dt= = Z Φ(k)−1dD Z <J(0,0)J(R, t)> eik·RdDrdDr0dt, (9.10) where the use of homogeneity allows the change of integration vari- ables r→R=r-r’,(−∞< R < r) andJ=J{v} is the correspond- ing nonlinear terms in physical space, e.g., in (8.2). Let us consider the ultraviolet limitk→ ∞while keepingk·R∼1. Let us make an assumption concerning the correlator: <J(0,0)·J(R, t)>. First of all we omit the pressure gradient potential part of the coupling and
leave only the solenoidal part, J0 =v×ω, since this is the impor- tant one for forming turbulence structure and generating vorticity. The second and more substantial assumption is that the correlator
<J0(0,0)·J0(R, t)>can be factorized as:
<J(0,0)·J(R, t)>=
=<v(0,0)·v(R, t >< ω(R, t)>< sinα(0)sinα(R, t)> . (9.11) This is an implicit assumption that the fastest time correlations cre- ated by the nonlinear coupling are the phase correlations and they control and balance the nonlinear interactions.
It is noted that the correlator <J0(0,0)·J0(R, t)>has relation with the correlator describing the fluctuations of helicityφ(rφ(r+δr
introduced in (5.18). Indeed, if the helicity correlators are anoma- lously large at small separation scales due to the alignment ofvandω
in helical sub-domains this would mean the corresponding reduction of<J(0,0)·J(R, t)>in these sub-domains due to the alignment. In difference to the local in physical space interrelation between v×ω
andv·Ω in the Navier-Stokes equations in the functional integral rep- resentation the global, non-local nature of helicity fluctuations enters naturally.
The separation of the amplitudes of velocity and vorticity in (9.11) is still possible because the velocity field is primarily determined by the large scales of order L and the vorticity field is primarily de- termined by the small scalesld and these two are uncoupled except
from the helicity related phase coherence that is accounted for by the correlator< sinα(0)sinα(R, t >.
With all the assumptions made we can now estimate the scaling power ofN L(k). We have:
[N L(k)] = [Φ−1] +D+ [<v(0,0)·v(R, t)>] + [< ω(0,0)·ω(R, t)>]+ +[< sinα(0)sinα(R, t)>]−2D−[dt] + [eik·R] =
= (y−D)−[dt] + [<v(0,0)·v(R, t)>] + [< ω(0,0)·ω(R, t)>]+
+[< sinα(0)sinα(R, t)>], (9.12)
where we took into account that[eik·R] = 0 and [dDr] = [dDR] =
−D. Let us consider the remaining terms above term by term. Con- sidering the expression (3.14) for the velocity correlation function
corresponding to the K41 spectrum it is clear that the leading con- tribution would be O(L2/3) and hence [< v(0,0)·v(R, t)>] = 0to
the leading order in powers of the Reynolds number.64 At the same
time [< ω(0,0)·ω(R, t)>] = [R−4/3= 4/3.
Consider first what happens if the K41 theory is accepted. Then apparentlyµ = 0, the correlation time ∆tK41 scales as [dt] =−2/3
andy =D. As far as the K41 theory is concerned there is no phase coherence, there are noBCCand theP df(v·ω/|v||ω|) is flat. This is to say that [sinα(0)sinα(R, t)>] = 0 for the time intervalt <∆tK41
and hence:
[<J0(0,0)·J0(R, t)>K41] =
= [<v(0,0)·v(R, t)>] + [< ω(0,0)·ω(R, t)>] = 4/3. (9.13) This can be interpreted as applying the quasi-Gaussian assumption to the basic correlator<J0(0,0)·J0(R, t)>.
Summing up together all the estimates above we arrive at the following:
[N L(k)]K41= 2/3, (9.14) or equivalently:
N L(k)K41∼(L/l)2/3=Re(k)1/2→ ∞. (9.15) Clearly this is not a satisfactory result. In the language of the pertur- bation theories when the functional integral is calculated by expan- sion in powers ofN L(k) the (9.15) would mean an unrenormalizable perturbation theory with each next order term growing as integer powers of large scale or the Reynolds number, i.e., like (L2/3)n.r
But in NT the situation is totally different. In this case we have from (9.9)y=D−µ/3 and [dt] =−2/3−µ/3. And now we suggest that due to the helicity phase and spatial coherence:
< sinα(0)sinα(R, t)>=λren2(R)∝λren2(k)∝k−2µ/3∝Re(k)−µ/2,
(9.16) withλren2(k) given by (8.60). Hence it is explicitly asserted here that the reduction of the coupling constant is a result of helicity coherence
64It is remainded thatRe→ ∞and this can if any of the three parameters, L, νL, or ν are set respectively as tending to∞or 0. Therefore, for instance,
O(L2/3) =O(L2/3/l2/3
d =O(Re
that is growing with the growth of wavenumbers or the corresponding decreasing of separation scales. In the ultraviolet limit k→ ∞ the helical fluctuations asymptotically tend to form Beltrami cells with extreme alignment. The subsequent singular factork−µ/3 in the in-
verse ”temperature” Φ−1 and the same k−µ/3 factor in dt give an
additional factor∝k−2/3 ∼Re(k)−µ/2. Altogether we obtain a nat- ural coupling reduction factorRe−µ =Re−1/2. It should be pointed out that the phase coherence (9.16) holds only for a short correla- tion time determined by the inverse of the typical frequency given by the scaling exponent (8.62). This frequency and the related eddy viscosity (8.63) are large by comparison with the K41 typical values, as was previously discussed. What it means is that the Beltrami like helical fluctuations of opposite sign exist only on a short time scale tending to zero in the limit Re → ∞. The fluctuations are virtual in this sense. As was explained previously (in the Foreword and Sec- tions 5, 8) the Beltrami cells of opposite sign either live only short life in much of the volume, or become stable only when their total volume is a diminutive fraction of the flow domain, asymptotically a fractal sub-domain and as a compensation the amplitudes of the velocity and vorticity fields are large (see Endnote s for discussion). In fact the velocity field inside the fractalBCCcan be estimated as:
vsing∼L1/6l1/6, (9.17)
instead of vreg ∼ l1/3. Altogether the helicityphase coherence fur-
nishes the required small parameter (kL)−2µ/3 → Re−µ = Re−0.5
that is needed for the convergence of the nonlinear coupling in the functional integral representation, or equivalently the asymptotic con- vergence of the RNG perturbation theory developed in Section 7. This convergence occurs for the unique value of µ= 0.5 as was de- rived in Section 7. Finally we obtain:
[N L(k)] = [BL(k)] = 0. (9.18) The results of this sub-section should be seen as to some extent inde- pendent, mathematically speaking at least, confirmation of the results of Section 8. It should be noted that all the asymptotic results above were obtained with logarithmic accuracy and disregard possible cor- recting factors of order {lnRe}x. Still such factors may be relevant
and will be discussed below. Let us analyze now the viscous terms
V(k).s
9.5 Energy spectrum in the buffer zone and skewness.
Is there anything that can be said based on NT about the energy spectrumE(k) in thebuffer zone? The point is that it is necessary to match the solution in the inertial range with the viscous dissipation range solution. In K41 theory this is an easy thing to do. The K41 eddy viscosity in the inertial range (3.34) is such that it matches naturally the molecular viscosity at k =kd i.e., K41 spectrum and
the exponentz = 2/3 that corresponds to the eddy viscosity (3.34) (and of course y = 3), together satisfy the general scaling solution conditions (7.1). On the other hand the solution (8.62) that was obtained in Section 8 and has been reconfirmed in the preceding sub- section 9.4 just now, the same K41 spectrum butz= 2/3+µ/3 = 5/6, do not satisfy the scaling conditions (7.1). As was pointed out in Section 8, the solution (8.62) is not really a scaling solution of the Navier-Stokes equations. At the same time this solution remains of course a scaling solution of the Euler inviscid equations. And now we should find a way to match this solution in the inertial range with a solution in the buffer zone where the viscous terms are of the same order as the nonlinear coupling. But the matching will not be smooth and easy now as it was for K41 theory in Section 5 above.
It was asserted in Levich (1987) that E(k) in the buffer zone should have an excess of energy over what it would have been if the spectrum was K41 all the way up tokd, as is effectively assumed in
K41 theory. In other words it was asserted that E)k) should have a flatter than the K41 slope somewhere in this range outside of the inertial range as it is redefined in (9.3).This can be deduced from the global matching of contributions from the nonlinear coupling term with that from the buffer zone and viscous range. Let us show how one can do this matching directly from the functional integral repre- sentation.
In accordance with (8.62) the nonlinear coupling generates the eddy viscosity in excess of the K41 eddy viscosity in the ultraviolet sub-range in k-space. This eddy viscosity terms should be somehow matched with the molecular viscosity terms. In particular the av-
eraged global contribution calculated for typical realizations of tur- bulence coming from the viscous terms should be the same order of magnitude, with logarithmic accuracy, that the contribution from