LÍMITE SUPERIOR
4.1 INGRESOS BRUTOS
Consider the velocity field inside a cube of edgelength 1/2πand its projection onto a finite cubic lattice inside that cube, for instance, a 3D grid withN equidistant lattice points in each direction. The
simplest boundary conditions would be the periodic ones:
vi(x, y, z) =vi(x+ 2πm, y+ 2πn, z+ 2πl), (4.1)
wherem, n, l are integers. Denote the coordinates of the grid points byr(n)wherenis a vector whose componentsn
iare integers from 1
toN and:
xi=ni/2πN. (4.2)
The velocity at a pointr(n)can be expanded in a triple Fourier series
as follows:
v(r(m)) = X
{k(n)}
v(k(n))ei(k(n)·r(n), (4.3)
wherek(m) is a vector with componentski=ni such that:
(k(n)·r(m)) =nimi/2πN, (4.4) and where: X {k(n)} = N X ni,nj,nl . (4.5)
The transformation inverse to (4.3) is:
v(k(n)) =N−3 X {r(n)} r(k(n))ei(k(n)·r(n), (4.6) where: X {r(n)} = N X ni,nj,nl . (4.7)
The shell averaged energy spectrum hence is:
Es(k(n)= 1/2 X
k(n)
/k(n)
(v(k(n))·v(−k(n))), (4.8)
where the summation is over all the orientations of the unit vector
energy spectrum itself is shell averaged energy spectrum. This is as- sociated with the energy spectrumE(k). Using the Parceval theorem it follows: E= X {k(n)} Es(k(n)) = 1/2N3 X {r(n)} v(r(n))2, (4.9)
where the sum is over all possible lengths ofk(n). In the same manner we introduce the spectral tensor Fijs(k)(k(n)) which is the analogue ofFij(k) and, in particular, the shell averaged helicity spectrum:
Hs(k(n)= X
k(n)
/k(n)
(v(k(n))ω(−k(n))), (4.10)
which is the analogue ofH(k) for the continuous case.
If the number of grid pointsN3 is big enough, or in other words
the resolution of DNS is high enough the space averaging or the shell averaging in the Fourier space would adequately approximate the ensemble averaging. This is not, however, invariably the case. Since
N is finite there will always be a random deviation of the space and shell averages from the genuine ensemble averages. For instance as a result of the statistically independent fluctuations in Es(k(n) the mean square deviation from the shell and space averaging will be of order O(N−1/2). Of course the real dynamics of the Navier-Stokes equation may somewhat alter this estimate but this is not the issue now. What is important to emphasize is that although for most quantities the deviations would be very small they are never small for the quantities for which the ensemble mean is zero. For instance for the helicity and helicity spectrum the space averages never are the same as the genuine ensemble averages, which for mirror symmetric flow are identically zero. Although the real fluctuations of helicity are determined by the dynamics of the Navier-Stokes equations and mat be not statistically independent the issue is that quasi-ergodic hypothesis should be applied with caution. In this sense it is more reliable to consider many time realizations of forced BigBox flow.
The time realizations are either generated from many initial con- ditions or by considering forced turbulence. The forced turbulence means that the force F 6= 0 in Eqs. (1.1). The force can be arbi- trary, in particular a useful and popular choice is a random force,
but with one important constraint. It should force only certain large scale harmonics of the flow and not affect directly the small scales so that not to contaminate the naturally developing turbulence cascade. The force should be in other words similar to a paddle that we used as a way to stir the flow in a pan when the K41 model was discussed above. The arbitrariness in the way the forcing is chosen is based on our conviction that the ensuing turbulent flow does not depend in its main features on how turbulence is triggered; to be sure a principle superseding K 41 theory in generality and most probably surviving in the asymptotic limit ofRe→ ∞, even if the K41 theory itself is not complete. In DNS the force will result in a flow that when averaged over sufficient number of time realizations and the averaging over space in each time realizations would be an adequate approximation to the ensemble averaged flows with stable steady state features like the energy spectrum. If there are coherent sub-domains in such a flow they should survive this double averaging and show up unam- biguously. This all is especially relevant because intermittency and coherent sub-domains are located in very small parts of the fluid vol- ume, which in discretized flow description corresponds to sets with relatively few lattice pointsN3
F/N3<<1.
Let us calculate the ensemble mean square deviation of the shell averaged helicity spectrum. The ensemble mean<H(k)>=<Hs,(k)>
=0, where the superscripts are omitted for simplicity. But the shell averaged H(k)s 6= 0.30 Let us assume that the fluctuations ofH(k) are statistically independent. Then evidently we are seeking the fol- lowing quantity: σ2H=<[H(k)sh]2>=<[X k/k H(k)]2>=X k/k X k0 < H(k)H(k0)> . 4.11) Now we use the statistical independence ofH(k:
< H(k)H(k0)>=< H(k)2> δk, k0. (4.12) We find for the variance (4.11):
σH2 =
X
k/k
< H(k)2>. (4.13)
30When it is not confusing the superscripts for the wavenumbers will be omit-
It should be noted that the zero, or in practice weak correlations as- sumption is equivalent roughly to assuming the Gaussian distribution for the fluctuations. It is more proper to say that a quasi-Gaussian assumption is made, because the strictly Gaussian assumption would lead to zero interaction between the velocity harmonics and subse- quently no dynamics at all. Using the Eq. (3.17), which is of course the same for the discrete description we obtain the factorized expres- sion:
σ2HG=
X
k/k
4k2<[Rev(k)2][Imv(k)2]>< sin2α(k)>, (4.14)
where we have assumed that the angles betweenRev(k) andImv(k), actually the phases α(k), are statistically uncoupled from the abso- lute magnitudes of these vectors, an assumption subsequent to the assumption of statistical independence ofH(k) fluctuations. Also it is obvious that: < Rev(k)2>=< Imv(k)2>= 1/2< E(k)> . (4.15) Hence we obtain: σHG2 = 2k2X k/k < E(k)2>, (4.16)
where the additional subscript G means Gaussian assumption. Fi- nally assuming that statistically there is rotational isotropy making possible replacing the summationP
k/k by 4πk
2, one obtains for the
helicity spectrum variance:
σ2HG= 1/2π[(E(k)2]. (4.17)
This is a fairly remarkable result in that it is independent of the resolutionN. It shows that in any realization of turbulence the shell averaged spectrum fluctuates primarily within the following interval:
−Es(k)/√2π≤H(k)sh≤+Es(k)/√2π. (4.18) In the next section it will be explained that the assumption of statis- tical independence ofH(k) that is roughly equivalent to the assump- tion of statistical independence of the phases α(k, t) is inconsistent
with the dynamics of the Navier-Stokes equations and obstructs the energy cascade to small scales. It should be pointed out that even though K41 theory is not complete but the general concept of the energy transfer to and dissipation at small scales is undoubtedly a correct vision of turbulence. Thus it should be always in our mind for testing the validity of assumptions. If an assumption contradicts this basic tenet than it is definitely wrong. Since the assumption of statistically independent helicity harmonics fluctuations are not com- patible with the unimpeded flow of energy from large scales to the small ones than this assumption is wrong. If it is wrong it means that the helicity harmonics H(k) are coherent. This means in fact thatα(k) phases are coherent. It is this phase coherence that results immediately in intermittency in physical space. It is a very general property of phase coherence of fields in Fourier space that it trans- forms and shows as a certain bunching effect for the large amplitudes of this field in physical space, i.e., in general sense intermittency. Thus the result that we must anticipate is that the helicity fluctua- tions are connected with turbulence intermittency and with no such intermittency the unimpeded energy flow to high wavenumbers is not possible. It is obvious without repeating that all the formulas of the previous sections are correct for the discrete description with integrals and volumes substituted by the sums and number of lattice points. The natural condition for a good resolution of DNS is as follows:
N3>> Nmodes, (4.19)
whereNmodes is defined by (2.22). This condition shows what enor-
mous computational power would be required for faithful DNS of tur- bulent flows for large values of the Reynolds number. This illustrates the desperate need of geophysical community to have model equations instead of the real Navier-Stokes equations that would eliminate most of the degrees of freedom but at the same time do not ”throw a child out of the basin together with water”, speaking figuratively. Such models may be possible but in the first place there must be if a not an analytical theory, such as we were used to in other physical disci- plines of the past, then at least a good qualitative comprehension of what turbulence really is.
In what follows the discrete and continuous description will be used wherever convenient without further comments and considered
equivalent. Nevertheless, it should be emphasized that there is a prin- cipal and not only practical significance in the relations (2.22) and (4.20). What they tell us is that despite the fact that the fluid flow as described by the partial differential Navier-Stokes equations is a space and time continuous problem with generally infinite number of degrees of freedom turbulence is always a problem with finite number of degrees of freedom proportional to the finite phase volumekd3that only asymptotically tends to infinity together with the unbounded growth of the Reynolds number. In this there is clear similarity with other nonlinear systems with finite degrees of freedom and compli- cated chaotic dynamics and herewith lies hope that may be much smaller phase space and much smaller number of degrees of freedom would be sufficient to describe turbulence faithfully.