I Introduction
Velocity increments across specified separation distances are important theoretical objects in studies of three-dimensional turbulence Kolmogorov (1941a). Their properties have been explored in a large number of papers in the past or so years and the more important results are summarized in Frisch (1995); Sreenivasan and Antonia (1997); Ishihara et al. (2009); Benzi and Biferale (2015); Iyer et al. (2019a). Analogues of velocity increments have also found interesting applications in other fields such as fracture mechanics Vernède et al. (2015), optical waves Solli et al. (2007) and foreign exchange rates in financial markets Ghashghaie et al. (1996). Velocity increments in turbulence, and their analogues in other fields just illustrated, exhibit intense fluctuations, possess fat-tailed distributions and are typically not space-filling. The consensus of results is that the velocity increments depart from classical self-similarity that was assumed to prevail at the time of the seminal work in Kolmogorov (1941a); Obukhov (1941a, b); Kolmogorov (1941b); Heisenberg (1948); Weizsäcker (1948); Onsager (1949). However, there do exist occasional claims that departures from self-similarity are artifacts of finite Reynolds numbers, and so will vanish in the limit of very large Reynolds numbers under ideal circumstances Qian (1998); Arenas and Chorin (2006); McComb et al. (2014); Tang et al. (2019); Antonia et al. (2019). Part of the reason for these latter claims is that the ‘consensus results’ are often based on data at modest Reynolds numbers, or complicated by remnant anisotropies, or adopt Taylor’s hypothesis (using time traces as one dimensional longitudinal cuts through the three-dimensional field), or employ the so-called extended self-similarity (ESS) or its variants (i.e., plotting various moments of velocity increments against the third-order), etc. These factors introduce uncertainties which, though believed to be benign, are not quantifiable precisely, and so lead to occasional divergence of conclusions. But the problem is of tremendous significance to the theory of turbulence Monin and Yaglom (1975) to be left in this ambiguous and inconstant state.
We have recently accumulated large databases of homogeneous and isotropic turbulence in large periodic boxes (as large as grid points) with good resolution in both space and time Yakhot and Sreenivasan (2005); Donzis and Yeung (2010); Yeung et al. (2018), which can be used to assess the status of fundamental issues such as self-similarity, intermittency, and universality. There is an acceptable likelihood of reaching satisfactory conclusions because: The highest Taylor microscale Reynolds number of these data is , which appears to be high enough to expect decent scaling (for Eulerian quantities) without the need for ESS or its variants; there is obviously no need for Taylor’s hypothesis because the data are spatial; we successfully remove by spherical averaging the residual anisotropies inherited from forcing and the cubic shape of the simulations box; and the amount of statistically stationary data available is adequate for high-order moments of velocity increments to converge reliably. Indeed, the analysis of the data shows that the departures from the estimates based on self-similarity assumptions on velocity increments are indeed real and do not vanish with increasing Reynolds number. They also show that, even in isotropic turbulence, there is a persistent difference between the longitudinal and transverse velocity increments, as was pointed out already in Chen et al. (1997); Dhruva et al. (1997); the particular new result, which we believe is of far-reaching theoretical consequence, is that the scaling exponents of the transverse increments saturate for higher order increment moments reminiscent of Burgers turbulence and passive scalars Gotoh (1994); Celani et al. (2000); Falkovich et al. (2001); Staicu and van de Water (2003); Iyer et al. (2018). We will briefly examine the reason why. The longitudinal moments might also saturate, but, if that does happen, it would do so for moments of far higher order—the ones that cannot be computed reliably.
Section II presents an account of numerical methods and flow parameters, as well definitions for later use. The account of numerical methods is necessarily brief because a more detailed description can be found in Refs. Yeung et al. (2012, 2015). Section III presents the bulk of the results and is followed by discussions and conclusions in Sec. IV.
Ii Numerical method, flow parameters and definitions
We solve the incompressible Navier-Stokes equations in three-dimensions,
(1) |
by using direct numerical simulations (DNS) on a triply periodic box with edge-length , where is the solenoidal velocity field (), is the kinematic viscosity and is the kinematic pressure, and the large-scale forcing in the range (technically, in the corresponding wavenumber range) Donzis and Yeung (2010). We use the standard pseudospectral scheme with exponential convergence, and calculate the nonlinear terms in physical space. The time-stepping is done with an explicit second-order Runge-Kutta integration to evolve the flow to a statistically stationary state to which all the present results correspond. The results have been averaged over a stationary period of at least large-eddy time scales , where is the so-called integral scale and is the root-mean-square velocity fluctuation. Both and are independent of viscosity or, equivalently, the Reynolds number. We present results in terms of the microscale Reynolds number , where the Taylor microscale is given by . The spatial resolution , where the grid spacing and the Kolmogorov scale , given in terms of the mean energy dissipation rate , is listed in Table 1, along with the other relevant flow parameters. In Appendix A we have verified that the results provided in this paper are consistent with those from shorter simulations Yeung et al. (2018) that used finer spatial and temporal resolution.
is a measure of the number of degrees of freedom in the three-dimensional field; and
is the ratio of the grid spacing to the Kolmogorov scale.We define a few parameters for later use in the paper. Consider the two-point velocity increment at location
across a separation vector
with magnitude , . Define the longitudinal increment , where is the unit vector along , and the transverse increment vector . The magnitudes are written, for the transverse case as an example, as . The velocity increment moment, also known as the structure function, at order , is defined as(2) |
where denotes space, time and angle (or spherical) averages Taylor et al. (2003); Iyer et al. (2017a). The angle averaging is performed to obtain the isotropic sector of from its SO(3) expansion Kurien and Sreenivasan (2001a); Biferale and Procaccia (2005). At sufficiently large , if there exists an inertial range of scales that are smaller than the integral scale (where energy injection occurs) and larger than the viscous scales (where dissipation manifests), the structure functions in that range are expected to scale as , where are the scaling exponents. For later use we note that the self-similar scaling attributed to Kolmogorov Kolmogorov (1941a) gives .
ii.1 Third order structure function
We shall first consider structure function of order , for which an exact result has been derived from Eq. 1 in the inertial range (if one exists). This so-called Kolmogorov’s -ths law Kolmogorov (1941b) is given by
(3) |
Figure 1 shows that, at , Eq. 3 is satisfied within error bars in the range , to both sides of which dissipative or large scale effects manifest to produce deviations from Eq. 3. The logarithmic local slope of given in the inset of Fig. 3 shows excellent agreement with the power-law exponent of unity in Eq. 3. The angle-averaged result shown in Fig. 1 corresponds to the isotropic sector of the SO(3) decomposition of and extends the inertial range by a factor of two over the Cartesian-averaged result for the same data (Fig. of Ref. Iyer et al. (2017b)). This dispels the explicit claim of Ref. Antonia et al. (2019) that the evidence for the 4/5-ths law does not exist.
Iii Results
iii.1 Second order structure function and the intermittency exponent
Before examining structure functions of order , it is instructive to test the incompressible relation in isotropic turbulence at scale ,
(4) |
Figure 2 displays , the ratio of the right hand side to the left hand side of Eq. 4, calculated along the three Cartesian directions and using the isotropic sector (filled circle) of the second order structure function. The remnant anisotropy from the cubic grid geometry and large scale forcing render velocity increments along different directions anisotropic to different degrees, resulting in different from unity even at moderately small scales. But the angle averaging, which retains only the isotropic sector, guarantees that at almost all scales, as the figure clearly shows. If the longitudinal and transverse second-order structure functions display power-law behaviors in the inertial range, Eq. 4 implies that the exponents and must be equal. The inset of Fig. 2 verifies this expectation.
Since the second-order exponents , as shown in the inset of Fig. 2, it suffices to examine more closely the longitudinal structure function (say), as is done in Fig. 3. The second-order structure function displays proper power-laws in the expected scale range (see inset of Fig. 2). The deviation of the exponent from the self-similar Kolmogorov exponent, obtained by least-squares, is plotted in Fig. 3(b) as a function of log-. The correction increases for lower but saturates at higher at a constant value of about 0.72, about higher than .
iii.2 Isotropy and fourth order quantities
Exact dynamical equations derived for isotropic structure functions of even orders Yakhot (2001); Hill (2001) contain mixed-order structure functions and structure functions of pressure and velocity increments. The equation for the longitudinal structure functions of order is given by
where and is the longitudinal pressure gradient structure function. Similar equations relating the transverse and mixed structure functions are also known Yakhot (2001); Hill (2001); Kurien and Sreenivasan (2001b). The pressure contributions were initially thought to be small in the inertial range Yakhot (2001); Kurien and Sreenivasan (2001b), which led to the result that the scaling exponents for a given order are equal, i.e., for .
In order to examine isotropy persuasively we substitute in Eq. III.2 to yield the following exact isotropic relation valid for :
(6) |
Here the large scale contributions drop out because . Figure 4(a) shows the ratio of the right hand side to the left hand side of Eq. 6, calculated along the three Cartesian directions and for the isotropic sector (filled circle) of the fourth order structure functions. The isotropic sector does indeed satisfy Eq. 6 beyond exceedingly well. This result ensures isotropy at order four for and for all .
We now examine the scaling exponents for order four and assess the contributions of pressure. Figure 4(b) compares the ratios and , which are essentially constant in the inertial range (approximately in the region ), suggesting that and display power-laws over this region. Also shown for comparison is the ratio , noting that due to incompressibility (see Fig. 2(b)). The exponents and show non-trivial differences in this range which indicate that the pressure contribution to Eq. 6 is not negligible — at least for this Reynolds number.
iii.3 Pressure contributions
In order to ascertain the role of pressure in Eq. III.2 we plot in Fig. 5 the ratio of the pressure term to that of the longitudinal structure function for orders and at ; here . With increasing order, the ratio increases in the intermediate scale range, with the gap between successive orders decreasing, perhaps suggesting that they level off to some non-zero value for some high orders not accessible to measurement today. A similar examination of ratios shows the same qualitative behavior. The conclusion is that the pressure effects between longitudinal and mixed structure functions increase upscale (see the green diamonds in Fig. 5) and with increasing order (though perhaps less rapidly), leading to the persistent differences observed in Fig. 4(b) between and . The inset compares the non-dimensional ratio at three different Reynolds numbers against scale normalized by , for purposes of examining the Reynolds number dependence of pressure contributions. The ratio decreases slowly with increasing Reynolds number causing the exponents to come closer Shen and Warhaft (2002); Iyer et al. (2017b). It follows that, for a given finite Reynolds number, the pressure contributions differentiate between the exponents and . On the other hand, it is well known that the pressure effect on the transverse structure functions is markedly smaller Kurien and Sreenivasan (2001b); Gotoh and Nakano (2003) and, in fact, decreases upscale, causing the mixed and the transverse exponents and to approach each other at smaller scales, as seen in Fig. 4(b).
iii.4 Saturation of higher order exponents
We summarize in Fig. 6 the exponents for integral orders. Focusing first on the longitudinal exponents , the data extend to for but had to be truncated at for for reasons of statistical convergence; a brief assessment of the convergence of moments is presented in the next section. The longitudinal data for the two Reynolds numbers shown agree with each other for , beyond which they begin to differ modestly; we cannot be conclusive that the differences represent genuine Reynolds-number effect and will not focus on that here. The longitudinal exponents appear to closely follow the model by Yakhot Yakhot (2001); Schumacher et al. (2007) (almost up to order ) while they increasingly deviate from the She-Leveque model She and Leveque (1994) and the -model Meneveau and Sreenivasan (1987) at higher orders. (The data differ also from an interesting model by Ruelle (2012) but we do not show this comparison here because the model does not preserve the concavity property of the exponents.)
Also shown in the figure are the even-order exponents for transverse structure functions. These exponents agree with the longitudinal data for moment orders approximately up to . Beyond that, for higher orders, focusing first on , the transverse exponents show a tendency to saturate to a value of about . For the higher , the tendency to saturate begins at lower moment order (perhaps ), and the saturation value appears to be about giving .
The saturation of transverse exponents suggests that there must be some huge excursions in transverse increments, unlike in the longitudinal counterparts, that imprint their characteristics on high-order structure functions. This is clearly seen in the traces provided in Fig. 7. In turbulence, the nonlinear effects that steepen gradients are balanced by the effect of pressure that mitigates it. What we have seen is that the pressure effect on transverse velocity increments is weaker, giving rise to steeper structures in those signals. Thus, transverse exponents saturate and exhibit a greater degree of intermittency than longitudinal increments Kraichnan (1991); Gotoh and Nakano (2003). Since the pressure effect seems to fade very slowly at a fixed order with increasing (see inset in Fig. 5), it is possible that the longitudinal exponents may also saturate in the limit . Indeed, Yakhot’s model predicts that even longitudinal exponents eventually saturate at as , but its verification is beyond the capabilities of the present data (or the foreseeable ones, since a simple extrapolation suggests that it would require Reynolds numbers beyond those occurring on Earth). What is certain from the -trend in Fig. 6 is that the higher order transverse exponents saturate.
iii.5 Statistical convergence of moments and tails of probability density functions
The statistical convergence of the higher order moments of the longitudinal and transverse increments is confirmed by the rapid decay towards both tails of the moment integrands as shown in Fig. 8. The integrands of moment orders for and are shown in Fig. 8, each at the lower end of the inertial range. The integrands peak well before the tail contributions decay, ensuring statistical convergence of the moments.
Saturation of transverse exponents at higher orders implies that present in the transverse velocity increments are jumps , which implies that the tails of the probability density function . Figure 9 verifies that this is indeed the case with collapsing for , across the inertial separations. In contrast, the compensated probability density functions of the longitudinal increments do not collapse in this fashion even for .
iii.6 Exponent derivatives
We now consider the derivatives of absolute values of structure functions of various orders. Define for longitudinal quantities the local slope of order- as
(7) |
whose constancy in the inertial range yields the longitudinal scaling exponent at order . Clearly,
, but the two may differ for odd orders. Differentiating Eq.
7 with respect to we get the exponent derivative(8) |
In particular, the exponent-derivative for is
(9) |
Figure 10 plots the zeroth-order derivative as a function of scale at three different Reynolds numbers. In the viscous limit all curves approach unity which one obtains using a Taylor series expansion. In the inertial range the curves approach a scale independent plateau which is the order derivative at . Intermittency exists even at order zero (as was noted already in Ref. Chen et al. (2005)) and is seen to saturate at for . In the multifractal model Parisi and Frisch (1985) this corresponds to the most probable Hölder exponent, at which the fractal set attains its maximum .
The exponent derivatives for for both the absolute longitudinal and even-order transverse increments at (Eq. 8) are summarized in Fig. 11. The derivatives decrease monotonically with the order since the exponents are concave in due to the Hölder inequality, with the longitudinal and transverse derivatives differing from order or so onward, with the latter dropping more steeply than the former. In the multifractal terminology, this means that the dominant Hölder exponent of the transverse increment is smaller than that of the longitudinal increment from order onwards rendering the transverse increments to be more intermittent than the longitudinal counterparts. The transverse order derivative reaches zero at order (consistent with the earlier finding about the saturation) whereas that of the longitudinal exponent continue to possess a positive slope which decreases with increasing order and roughly reaches the asymptotic of the She-Leveque model She and Leveque (1994). These conclusions are entirely consistent with Figs. 4(b) and 6. Since negative scaling exponents for incompressible flows are ruled out in the multifractal model Frisch (1995), we note that the most dominant Hölder exponent corresponding to the saturation of transverse exponents, which physically translates to the presence of shock-like structures in the transverse increment field, as seen in Fig. 7(d).
Iv Discussion and conclusions
At the time of the seminal works of Kolmogorov Kolmogorov (1941c, b); Obukhov (1941a, b); Heisenberg (1948); Weizsäcker (1948); Onsager (1949), it was thought that fluid turbulence comprises scales that progressively degenerate in structural integrity, essentially becoming more isotropic as the energy cascade proceeds. On the basis of this scale invariance the second order velocity structure function was deduced to scale as — or, equivalently, the energy spectrum as , where is the wavenumber. Its convincing experimental verification Grant et al. (1962) had to wait many years, which also sowed the seeds for the incompleteness of the self-similar theory. Subsequently a number of works starting from Kolmogorov (1962) have revised this earlier picture and it is now well known that turbulent structures tend be less space filling and intermittent, far from being spherical, causing departures from self-similarity Anselmet et al. (1984); Benzi et al. (1995); Herweijer and van de Water (1995); Sreenivasan and Dhruva (1998); Gotoh et al. (2002); Schumacher et al. (2007); Saw et al. (2018). However, a few workers in the recent past Tang et al. (2019); Antonia et al. (2019) have wondered if the deviations from self-similarity should be attributed to insufficiently high Reynolds number, and have particularly noted that many measurements and simulations do not satisfy the -ths law convincingly enough.
Using a DNS database that spans over a decade in with the highest , and by establishing statistical isotropy by decomposing the structure functions in the SO(3) basis, we have shown that the -ths law holds to within statistical errors (Fig. 1). We have explicitly demonstrated that the second order exponents depart from the Kolmogorov value of and approach a constant value of at higher Reynolds numbers. This result soundly demonstrates (the small effect of) intermittency even at the level of the energy spectrum. The fact that the second-order inertial range exponent possesses a constant positive value which initially increases with the Reynolds number has important theoretical implications Drivas and Nguyen (2019).
We have obtained further results. We have confirmed under convincing conditions that intermittency increases with increasing order for longitudinal, transverse and mixed structure functions, but shown, quite importantly, that the transverse exponents differ from longitudinal ones for orders greater than about . This result was already obtained Chen et al. (1997); Dhruva et al. (1997), but was called into question from symmetry arguments L’vov et al. (1997) by stating that longitudinal and transverse structure functions mix different scaling functions and may obscure pure scaling. Be that as it may, the present results show convincingly that they are different when they scale. This result suggests the very notion of scaling in turbulence is more complex than traditionally thought.
Lastly, perhaps the most important among the new results, is the result that the transverse exponents saturate for large moment orders. If there is a mixing of scaling functions, the transverse exponents will control the scaling for very small scales (effectively, very large Reynolds numbers), so the result is fundamentally important. For the highest Reynolds number we have been able to consider, ; this is the fractal co-dimension Frisch (1995) of the cliffs in the velocity field in the multifractal picture, suggesting that large jumps occur in the field of transverse velocity increments. A consistent physical picture is the likely prevalence of vortex sheets rather than more complex singularities. This is the first such confirmation in the general case of homogeneous isotropic turbulence, thus raising the possibility that saturation of velocity exponents is an endemic feature of turbulence at all high Reynolds number flows. If this were indeed the case, the phenomenological small-scale models will have to account for their explicit presence.
V Acknowledgments
We are grateful to many of our colleagues with whom we have discussed these results over the years, in one form or another, and to Xiaomeng Zhai and Matthew Clay for their participation in the computations. This work is partially supported by the National Science Foundation (NSF), via Grants No. ACI- and No. ACI- at the Georgia Institute of Technology. The computations were performed using supercomputing resources provided through the XSEDE consortium (which is funded by NSF) at the Texas Advanced Computing Center at the University of Texas (Austin), and the Blue Waters Project at the National Center for Supercomputing Applications at the University of Illinois (Urbana-Champaign).
Appendix A Numerical resolution
In order to examine the effects of finite spatial and temporal resolution of the DNS we have compared the velocity structure functions from simulations with two different resolutions up to order at . The finer DNS has a spatial resolution of almost half the Kolmogorov length scale and a time-step which is of the Kolmogorov timescale Yeung et al. (2018). In comparison, the coarser DNS has a grid spacing of almost and a time-step which is of . Figure 12 compares the longitudinal and transverse structure functions from two simulations at order at . In the range , the exponents obtained from a Taylor-series expansion show that only the structure functions from the finer DNS are reliable. Notwithstanding the small- result, in the inertial range, the structure functions from both simulations show excellent agreement, as seen in Fig. 12, with the same inertial range exponents obtained from least-square fits.
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