Decaying Turbulence

K.R. Sreenivasan

# Laws of turbulence decay from direct numerical simulations

John Panickacheril John1 Diego A Donzis2 and Katepalli R Sreenivasan 3
1Department of Mechanical and Aerospace Engineering, New York University, New
York, NY 10012, USA
2Department of Aerospace Engineering, Texas A&M University, College Station,
TX 77843, USA
3Department of Mechanical and Aerospace Engineering, Department of Physics and
Courant Institute of Mathematical Sciences, New York University, New York, NY
10012, USA<EMAIL_ADDRESS>
###### Abstract

Inspection of available data on the decay exponent for the kinetic energy of
homogeneous and isotropic turbulence (HIT) shows that it varies by as much as
100%. Measurements and simulations often show no correspondence with
theoretical arguments, which are themselves varied. This situation is
unsatisfactory given that HIT is a building block of turbulence theory and
modeling. We take recourse to a large base of direct numerical simulations and
study decaying HIT for a variety of initial conditions. We show that the
Kolmogorov decay exponent and the Birkhoff-Saffman decay are both readily
observed, albeit approximately, for long periods of time if the initial
conditions are appropriately arranged. We also present, for both cases, other
turbulent statistics such as the velocity derivative skewness, energy spectra
and dissipation, and show that the decay and growth laws are approximately as
expected theoretically, though the wavenumber spectrum near the origin begins
to change relatively quickly, suggesting that the invariants do not strictly
exist. We comment briefly on why the decay exponent has varied so widely in
past experiments and simulations.

###### keywords:

decaying turbulence, scaling laws, universality, incompressible turbulence

##  1 Introduction

Homogeneous and isotropic turbulence (HIT) has inspired many theoretical
studies of turbulence (see, e.g., Refs. [1, 2, 3]), and provided the basic
modeling constant on energy dissipation [4]. It is the simplest turbulent flow
possible and its conceptually minimalistic dynamics consists of a monotonic
decay of turbulent kinetic energy, with a resulting rearrangement of the
energy spectral density. HIT has been studied for more than sixty years but
remains somewhat murky even today with basic contradictions and unresolved
anomalies. Our goal is to address them by using large direct numerical
simulations (DNS) as the tool. This approach allows us a degree of control on
initial conditions that is not attained in grid turbulence experiments.

Once HIT is produced in some manner (experimentally by sweeping a grid of bars
or by pushing a flow past the grid, or, computationally by injecting initial
energy into some chosen wave number bands), the equation for the average
turbulent kinetic energy reduces to [1]

$\frac{dk}{dt}=-\langle\epsilon\rangle,$ (1)

where the turbulent kinetic energy $k=\frac{3}{2}u^{\prime 2}$,
$u^{\prime}=\sqrt{\langle u_{i}u_{i}\rangle/3}$ being the root-mean-square
velocity, and the average rate of turbulent energy dissipation
$\langle\epsilon\rangle\equiv\nu\langle(\partial u_{i}/\partial x_{j}+\partial
u_{j}/\partial x_{i})^{2}\rangle)$. Here, summation is implied over the
repeated index, and $u_{i}$ is the velocity component in the direction
$x_{i}$. This simple equation cannot be solved because
$\langle\epsilon\rangle$ is not known in terms of $k$ (or $u_{i}$).

It is traditional to think that, after some initial adjustment period, the
energy decays as a power law in time $t$ as

$k=Kt^{-n},$ (2)

where $n$ is a universal constant and $K$ has the dimensions of energy
depending on the energy initially injected, its spectral distribution, etc.,
and $t$ is suitably normalized. The focus of attention is the decay exponent
$n$. This power-law behavior is assumed to be valid at high enough Reynolds
numbers; so we exclude from consideration very large $t$ for which the energy
has plummeted to such low levels that the Reynolds number becomes very small.
A similar relation for the length scale $L$ of turbulence is

$L=L_{0}t^{m},$ (3)

where $L_{0}$ is some system-specific length and the exponent $m$ is thought
to be universal. Typically, $L$ is taken as the integral scale defined as
$\left(3\pi/4\right)\left(\int_{0}^{\infty}\kappa^{-1}E(\kappa)d\kappa\right)/\left(\int_{0}^{\infty}E(\kappa)d\kappa\right)$,
where $E(\kappa)$ is the energy spectrum, and is the same as the integral of
the longitudinal correlation function in real space. One expects, because of
nonlinear interactions among the large degrees of freedom of the flow, that
HIT at high Reynolds numbers evolves into an asymptotic universal state that
is independent of initial conditions, i.e., $n$ and $m$ assume universal
values.

Despite the simplicity of decaying turbulence, the universality of the decay
exponents has not been confirmed convincingly. We present in figure 1 two
histograms of the decay exponent $n$ from the literature, which show a wide
spread uncharacteristic of universal quantities. (A similar collection for $m$
is less useful because fewer studies have measured the length scale $L$.)
Figure (a) includes data from grid experiments in wind tunnels, for which the
flow far enough away from the grid is thought to be nearly homogeneous and
isotropic, as well as pull-through grids in liquid tanks for which HIT is
similarly attained after a certain initial adjustment time. Figure 1 (b) shows
data from simulations. Together the two figures demonstrate that the state of
decay depends, presumably, on many factors such as the Reynolds numbers,
initial and residual anisotropies, boundary effects, measurements being made
too close to the grid, the quality of the grid itself, the ubiquitous use of
virtual origin in plotting data, inadequate ranges over which power laws are
fitted, unsatisfactory simulation parameters, etc. It is clear that the
problem demands closer attention.

(a) (b)

Figure 1:  Histogram of decay exponents from the literature: (a) experiments,
(b) simulations. Data in (a) are from I [5], II [6, 7, 8, 9], III [10], IV
[11, 12, 13], V [14, 15, 16, 17, 18], VI [19], VII [20], VIII [21], IX [22,
23, 24]. In (b), the initial spectrum with
$E\left(\kappa\right)\propto\kappa^{4}$ near the origin correspond to:
$\kappa^{4}$-I [25, 26, 27], $\kappa^{4}$-II [28], $\kappa^{4}$-III [29],
$\kappa^{4}$-IV [30, 31, 32, 33], $\kappa^{4}$-V [34]. The initial spectrum
with $E\left(\kappa\right)\propto\kappa^{2}$ for small $\kappa$ corresponds to
cases: $\kappa^{2}$-I [35, 36], $\kappa^{2}$-II [27], $\kappa^{2}$-III [29],
$\kappa^{2}$-IV [31, 32, 37]. For the case [33], the simulations were
compressible. Simulations here include DNS, LES using different numerical
methods, and EDQNM. Some experiments and simulations are no doubt more
thorough than others, but we cannot a priori discard any of them on the basis
of available information. It should not be inferred that the “correct"
exponent is necessarily the most frequently observed one.

After a brief discussion of the basic theoretical framework for decaying
turbulence, we discuss numerical simulations in section 3, present results in
sections 4 and 5, and close with a brief discussion and summary of results in
section 6.

##  2 Theoretical framework

One natural step in closing the equation at high Reynolds numbers is to
represent $\langle\epsilon\rangle$ in terms of the energy introduced into the
turbulent system by invoking the so-called zero-th law of turbulence: the
energy injection rate at large scales equals the dissipation rate at small
scales. We might take $u^{\prime}$ and the integral scale $L$ as the velocity
and length scales characteristic of energy injection. Writing

$\langle\epsilon\rangle=C_{\epsilon}u^{\prime 3}/L,$ (4)

where $C_{\epsilon}$ is a constant of the order unity111$C_{\epsilon}$ depends
on the Reynolds numbers when the Reynolds number is small. This dependence is
reasonably well understood; see Refs. [38, 39, 40, 41]. Even in the high
Reynolds number limit $C_{\epsilon}$ depends modestly on initial conditions
[38]., (1.1) becomes

$\frac{3}{2}\frac{du^{\prime 2}}{dt}=-C_{\epsilon}u^{\prime 3}/L,$ (5)

and can be solved readily if we specify the growth of the length scale. As one
point of reference, for a confined apparatus in which the length scale
saturates at the size of the apparatus containing the turbulence (this could
be the size of the computational box in DNS studies), we can solve (5) to
obtain

$k\sim t^{-2}\hskip 56.9055ptL\sim t^{0}.$ (6)

There are experiments that confirm this result [41, 42, 43]. In particular,
the towed-grid experiments in a water tank [41] showed the early part of the
decay (50 < tU/M < 1,000) yielded an $n$ of 1.1 whereas, as the tank walls
started to inhibit the growth of the integral scale, $n$ grew to 1.5 for 1,000
< tU/M < 10,000); here $t$ is measured from the time that the grid of mesh
size $M$ is pulled with a velocity $U$ through the observational point.

As another point of reference, the appealing idea of self-similarity [1]
yields

$k\sim t^{-1}\hskip 56.9055ptL\sim t^{1/2}.$ (7)

Relations (2.3) and (2.4) happen to roughly bound the extreme values observed
in experiments and simulations.

Other power laws can be deduced on the assumption that certain invariants of
the Kármán-Howarth equations exist. The most important invariants are related
to the spectral behavior near the wavenumber origin (or the behavior of
correlation functions for large $r$). We may write the three-dimensional
energy spectrum at low wavenumbers in the form [1]

$E(\kappa,t)=\frac{2}{\pi}I_{S}(t)\kappa^{2}+\frac{1}{3\pi}I_{L}(t)\kappa^{4}+.....,$
(8a) where $I_{S}$ and $I_{L}$, known respectively as the Birkhoff-Saffman
[44, 45] and Loitsiansky [46] invariants, are defined as $I_{BS}=u^{\prime
2}\int_{0}^{\infty}r^{2}\frac{1}{2r^{2}}\frac{\partial}{\partial
r}\left[r^{3}f(r,t)\right]dr$ (8b) and $I_{L}=u^{\prime
2}\int_{0}^{\infty}r^{4}f(r,t)dr.$ (8c)

In (8b) and (8c), $f$ is the longitudinal correlation function that depends
solely on the scalar value of the separation distance, $r$. Two classical
theories for decaying turbulence based on the existence of one or the
invariants have been proposed. If $I_{BS}=0$ [1], we have
$E\left(\mathbf{\kappa}\right)\propto\kappa^{4}$ near the origin; in this
case, $I_{L}$ is the invariant of motion, interpreted as the conservation of
angular momentum [47]. However, for certain initial conditions where the
correlation does not decay faster than $r^{-3}$ at large scales, Birkhoff [44]
and Saffman [45] independently argued that $I_{L}$ diverges and $I_{BS}$ is
the invariant of motion. This corresponds to the case
$E\left(\kappa\right)\propto\kappa^{2}$ at the origin, and is interpreted as a
result of conservation of linear momentum [47].

These constraints on the dynamical motion of turbulence can be exploited to
obtain the power law relation for $u^{\prime 2}$ and $L$. The existence of
these invariants, can be shown [44, 45, 48, 10]) to lead, in each case, to:

Birkhoff-Saffman: $\displaystyle\ \ n=6/5,\ \ m=2/5$ and Loitsiansky:
$\displaystyle\ \ n=10/7,m=2/7.$

There is no obvious way to know if any of these constraints applies a priori
to given experimental conditions. As a practical matter, the integrals in
(2.5b) and (2.5c) can extend only as far the size of the apparatus. Looking at
the histogram in figure 1, one comes to the conclusion that the most likely
decay exponent may be about 1.2 in both experiment and simulations, but no
convincing effort has been made in those instances to relate that exponent to
the existence of the Birkhoff-Saffman invariant. Thus, it is unclear at this
point if there is a universal power law exponent (indeed, whether there exists
a power law), and if the theory based on invariants plays a role in the
experiment.

On the other hand, in numerical simulations, one can impose either
$E(\kappa)\propto\kappa^{2}\textrm{ or }\kappa^{4}$ at low wavenumbers
(limited, of course, by the inverse box size) as the initial condition for the
energy spectrum. In such cases, the state of turbulence may well be
constrained by the invariants due to the specified spectral shape at low
wavenumbers, and one may explore whether the decay exponent $n$ turns out to
be $10/7$ or $6/5$, depending on whether $E(\kappa)\propto\kappa^{4}$ or
$E(\kappa)\propto\kappa^{2}$ near the origin. This is a specific thrust of the
present work.

Figure 2:  Initial spectra. (1) Dashed black line: energy spectra from fully
developed forced turbulence simulation; (2) Blue line: modified initial energy
spectra with $\kappa^{2}$ scaling and $\kappa_{p}=21$; (3) Red line: modified
initial energy spectra with $\kappa^{2}$ scaling and $\kappa_{p}=50$,
translated upwards to increase initial Reynolds number. (4) Green line:
modified initial energy spectra with $\kappa^{4}$ scaling and $\kappa_{p}=21$;
(5) Magenta line: modified initial energy spectra with $\kappa^{4}$ scaling
and $\kappa_{p}=50$ translated upwards to increase the initial Reynolds
number. See text for explanations.

##  3 Numerical simulations and turbulence parameters

We conduct direct numerical simulations of turbulence in periodic boxes for a
wide variety of initial conditions. The code uses a pseudospectral formulation
in space and a second-order Runge-Kutta scheme for time integration. In all
simulations, we generate random Gaussian-distributed velocity field as initial
conditions. In DNS studies of decaying turbulence [35, 25, 26], it is
customary to use an initial energy spectrum of the form,
$E(\kappa)\propto\left(\kappa/\kappa_{p}\right)^{q}\exp\left[-\left(\kappa/\kappa_{p}\right)^{2}\right]$
with the power law of $q=2\textrm{ or }4$ dominating low wave numbers,
$\kappa<\kappa_{p}$. In this work, we modify the energy spectra of forced
fully developed turbulence at low wave numbers such that, for
$\kappa<\kappa_{p}$, the initial energy spectrum has the form
$E\left(\kappa\right)\propto\kappa^{q}$, where $q=2\textrm{ or }4$. For
$\kappa>\kappa_{p}$, the initial energy spectrum has the same form as that of
fully developed turbulence in forced simulations. The spectrum is adjusted to
be continuous at $\kappa=\kappa_{p}$. This modification results in lower
initial Reynolds number compared to the forced stationary case. To increase
the initial Reynolds number in some cases, we translated the modified entire
energy spectrum upwards such that the initial Taylor Reynolds number is
nominally comparable to that in forced stationary case in the same box. (For
reference, the Taylor microscale Reynolds number
$R_{\lambda}=u^{\prime}\lambda/\nu$ where
$\lambda=\sqrt{u^{{}^{\prime}2}/\langle(\partial u/\partial x)^{2}}\rangle$,
and $\nu$ is the kinematic viscosity.) A few initial energy spectra along with
an example of the stationary case are shown in figure 2.

The relative value of $k_{p}$ is an important factor in determining the
behaviour of energy decay; in our study, it has the range $5-80$ in units of
the smallest allowed wave number $\kappa_{min}=2\pi/L_{box}$, which is unity
for all simulations. For all cases, the integral length scale remains smaller
than about 10% of the box size. Simulations were made on grid sizes ranging
from $256^{3}\textrm{ to }1024^{3}$, resulting in $R_{\lambda}$ ranging from
23 to 456. The simulation conditions are summarized in table 1.

| $\mathbf{\kappa^{4}\textrm{ cases }}$ |  |
---|---|---|---
Cases | $R_{\lambda,0}$ | $R_{L,0}$ | $R_{L,0}/R_{\lambda,0}^{2}$ | $\kappa_{p}$ | $\kappa_{b}$ | $\kappa_{p}/\kappa_{b}$ | $\kappa_{max}$ | $\nu$ | $L_{0}$ | $N$ |
1.1 | 23 | 33 | 0.062 | 10 | 11.71 | 0.85 | 127 | 0.0031 | 0.154 | 256 |
1.2 | 60 | 78 | 0.022 | 21 | 11.71 | 1.79 | 127 | 0.0031 | 0.099 | 256 |
1.3 | 65 | 168 | 0.039 | 5 | 24.27 | 0.21 | 255 | 0.0011 | 0.221 | 512 |
1.4 | 43 | 83 | 0.045 | 10 | 24.27 | 0.41 | 255 | 0.0011 | 0.135 | 512 |
1.5 | 25 | 39 | 0.062 | 21 | 24.27 | 0.87 | 255 | 0.0011 | 0.085 | 512 |
1.6 | 180 | 335 | 0.010 | 10 | 24.27 | 0.41 | 255 | 0.0011 | 0.127 | 512 |
1.7 | 155 | 242 | 0.010 | 21 | 24.27 | 0.87 | 255 | 0.0011 | 0.084 | 512 |
1.8 | 142 | 183 | 0.009 | 50 | 24.27 | 2.06 | 255 | 0.0011 | 0.043 | 512 |
1.9 | 77 | 94 | 0.016 | 80 | 24.27 | 3.29 | 255 | 0.0011 | 0.029 | 512 |
1.10 | 9 | 7 | 0.168 | 50 | 24.27 | 2.06 | 511 | 0.0011 | 0.043 | 512 |
1.11 | 21 | 17 | 0.075 | 50 | 24.27 | 2.06 | 511 | 0.0011 | 0.043 | 512 |
1.12 | 45 | 87 | 0.043 | 21 | 48.47 | 0.43 | 511 | 0.0004 | 0.069 | 1024 |
1.13 | 70 | 184 | 0.038 | 10 | 48.47 | 0.21 | 511 | 0.0004 | 0.116 | 1024 |
1.14 | 422 | 1171 | 0.010 | 10 | 48.47 | 0.21 | 511 | 0.0004 | 0.127 | 1024 |
1.15 | 383 | 566 | 0.004 | 50 | 48.47 | 1.03 | 511 | 0.0004 | 0.037 | 1024 |
1.16∗ | 88 | 111 | 0.014 | 80 | 48.47 | 1.65 | 511 | 0.0002 | 0.031 | 1024 |
| $\mathbf{\kappa^{2}\textrm{ cases }}$ |  |
Cases | $R_{\lambda,0}$ | $R_{L,0}$ | $R_{L,0}/R_{\lambda,0}^{2}$ | $\kappa_{p}$ | $\kappa_{b}$ | $\kappa_{p}/\kappa_{b}$ | $\kappa_{max}$ | $\nu$ | $L_{0}$ | $N$ |
2.1 | 26 | 41 | 0.061 | 10 | 11.71 | 0.85 | 127 | 0.0031 | 0.181 | 256 |
2.2 | 71 | 101 | 0.020 | 21 | 11.71 | 1.79 | 127 | 0.0031 | 0.116 | 256 |
2.3 | 70 | 203 | 0.041 | 5 | 24.27 | 0.21 | 255 | 0.0011 | 0.257 | 512 |
2.4 | 46 | 101 | 0.048 | 10 | 24.27 | 0.41 | 255 | 0.0011 | 0.157 | 512 |
2.5 | 28 | 49 | 0.063 | 21 | 24.27 | 0.87 | 255 | 0.0011 | 0.099 | 512 |
2.6 | 197 | 424 | 0.011 | 10 | 24.27 | 0.41 | 255 | 0.0011 | 0.157 | 512 |
2.7 | 175 | 304 | 0.010 | 21 | 24.27 | 0.87 | 255 | 0.0011 | 0.099 | 512 |
2.8 | 170 | 241 | 0.008 | 50 | 24.27 | 2.06 | 255 | 0.0011 | 0.051 | 512 |
2.9 | 95 | 127 | 0.014 | 80 | 24.27 | 3.29 | 255 | 0.0011 | 0.034 | 512 |
2.10 | 13 | 9 | 0.165 | 50 | 24.27 | 2.06 | 511 | 0.0011 | 0.051 | 512 |
2.11 | 28 | 20 | 0.070 | 50 | 24.27 | 2.06 | 511 | 0.0011 | 0.051 | 512 |
2.12 | 49 | 107 | 0.044 | 21 | 48.47 | 0.43 | 511 | 0.0004 | 0.081 | 1024 |
2.13 | 76 | 231 | 0.040 | 10 | 48.47 | 0.21 | 511 | 0.0004 | 0.140 | 1024 |
2.14 | 456 | 1476 | 0.007 | 10 | 48.47 | 0.21 | 511 | 0.0004 | 0.154 | 1024 |
2.15 | 436 | 716 | 0.004 | 50 | 48.47 | 1.03 | 511 | 0.0004 | 0.043 | 1024 |
2.16∗ | 373 | 535 | 0.004 | 80 | 48.47 | 1.65 | 511 | 0.0002 | 0.046 | 1024 |
Table 1: Table of simulation parameters and initial conditions. Here
$R_{\lambda,0}$ and $R_{L,0}$ are the initial Reynolds numbers based on the
Taylor microscale and integral scale, respectively; $\kappa_{p}$ is the
wavenumber up to which the initial spectrum has a prescribed power law of the
form $E\left(\kappa\right)\propto\kappa^{q}$ near the origin, where
$q=2\textrm{ or }4$. The parameter, $\kappa_{b}$ is the bottleneck wavenumber
based on forced turbulence (refer to the text for more details) and
$\kappa_{max}$ is the maximum wavenumber supported by the grid. Also, $\nu$
and $L_{0}$ are the kinematic viscosity of the fluid and initial integral
length of the flow. In all simulations, the lowest resolved wavenumber,
$\kappa_{min}$ is $1$; $N$ is the linear grid size. The number of realizations
$N(r)$ used to calculate the ensemble average is 5 for all cases, except for
1.14 and 2.16 for which it is 4. The starred $\left({}^{*}\right)$ cases 1.16
and 2.16 have the initial spectrum of the form
$E\left(\kappa\right)\propto\left(\kappa/\kappa_{p}\right)^{q}\exp\left[-\left(\kappa/\kappa_{p}\right)^{2}\right]$,
for both q = 2 and 4.

##  4 Preliminary results on decay exponents

In experiments behind grids in wind tunnels, for which the power-law extends
typically over a decade or so, researchers have often used by trial and error
a virtual origin to improve the quality of power law fits [6]. Since we
perform simulations for over a thousand integral scales, we calculate the
decay exponent assuming the existence of the power law decay of the form
$k\propto t^{-n}$ without invoking a virtual origin. The incorporation of a
modest virtual origin makes no difference to our conclusions on energy decay.
We obtain the scaling exponent locally by evaluating

$n=-\frac{d\ln k}{d\ln t}.$ (9)

A plateau implies the existence of a well-defined power law.

(a) (b) (a) (b)

Figure 3:  The decay exponent for all cases. Initially, (a)
$E\left(\kappa\right)\propto\kappa^{4}$ and (b)
$E\left(\kappa\right)\propto\kappa^{2}$ for small $\kappa$. The red dashed
lines correspond to classical theories: (a) $-10/7$, Kolmogorov’s form; (b)
$-6/5$, Birkhoff-Saffman form.

In our first attempt, we sampled a variety of initial conditions in terms of
Reynolds number and spectral form of turbulent energy, roughly mimicking the
experiment, and obtained decay exponents for all cases, as displayed in
figures 3 (a,b). Figures (a) and (b) correspond, respectively, to $\kappa^{4}$
and $\kappa^{2}$ behaviours at low wavenumbers. In both figures, time is
normalized by the initial eddy turnover time, $T_{e}=L_{0}/u^{\prime}_{0}$,
the ratio of initial values of the integral length scale to the root-mean-
square (rms) velocity. It is readily seen that not all cases have a plateau.
For those with approximate plateaus, the numerical values are not all the
same. Further, it seems that the plateau is sometimes achieved after a
relatively long time, $t/T_{e}\geq 0(100)$ (which often corresponds to the
outer edge of power-law fits in wind tunnel experiments). If one does not
observe the decay for long enough times and take local slopes, it appears that
the dip in early times can be construed as a power law. These slopes indeed
mimic the variability in the histogram (Figs. 1 (a) and (b)). It must also be
noted that not one of them has unity exponent (and the one case of $n$ of
about $2.25$ appears to correspond to the final period of decay).

Does this mean that there is no well-defined decay exponent? It suggests,
instead, that better control of conditions may be needed to be more
definitive. First, we need the Reynolds numbers to be high enough during the
decay process; since, for all exponents greater than unity, the Reynolds
number decreases with time, this condition is especially demanding for the
Kolmogorov case because of its larger decay exponent. We ought to make sure
that the energy input occurs at wavenumbes sufficiently far away from the the
dissipative range; otherwise, a substantial decay could occur directly at the
energy-containing scale itself. Further, the length scale should not grow to
be a substantial fraction of the computational box. These are often
conflicting requirements, and only a few runs of table 1 satisfy these
conditions, even if only approximately. We shall focus on just those cases
here. Particularly relevant is the role of the postulate on the permanence of
large eddies [46].

A quick examination of table 1 suggests that cases 1.6, 1.7, 1.8, 1.14 and
1.15 have the best chance of retaining high enough in Reynolds number for the
$\kappa^{4}$ cases. Likewise, 2.6, 2.7, 2.9, 2.14 and 2.15 appear to be the
most suitable among the $\kappa^{2}$ cases.

The second criterion is based on the fact that the energy-containing scales
must be distinct from the viscous-dominated scales of turbulence; in other
words, $\kappa_{p}$ should be far enough to the left of the dissipative
region. For box turbulence for which the lowest resolved wavenumber
$\kappa_{min}=1$, we stipulate as a guidance that $\kappa_{p}$ should satisfy

$\frac{\kappa_{p}}{\kappa_{b}}<1,$ (10)

where $\kappa_{b}$ is the wavenumber corresponding to the bottleneck region,
centered at $\kappa_{b}=0.13/\eta$ [49]. Here,
$\eta=\nu^{3}/\langle\epsilon_{i}\rangle^{1/4}$ is the Kolmogorov length scale
from forced turbulence simulations, which satisfies the criterion that
$\kappa_{max}\eta>1.5$ and resolves most significant scales in the problem;
$\langle\epsilon_{i}\rangle$ is the space-time averaged dissipation in the
stationary state prior to the onset of decay.

(a) (b)

Figure 4:  Spectra scaled on Kolmogorov variables for two typical $\kappa^{4}$
cases: (a) 1.6 and (b) 1.9. (a) is one of the high Reynolds number cases but
(b) is not. See table 1 and text for more details.

(a) (b)

Figure 5:  Spectra scaled on Kolmogorov variables for two typical $\kappa^{4}$
cases: (a) 2.7 and (b) 2.4. (a) is one of the three high Reynolds number cases
but (b) is not. See table 1 and text for more details.

If we apply this additional criterion, it is immediately clear that 1.6, 1.7
and 1.14, as well as 2.6, 2.7 and 2.14, are the probably the best cases to
consider. We will consider mostly these cases and comment on others only when
appropriate.

One general comment is useful before considering specific results. Figures 4
and 5 show the energy spectra normalized according to the Kolmogorov form for
small scales. For both cases, after some initial transients, the higher-
wavenumber spectra collapse to the standard accuracy (see [46]) under
Kolmogorov’s [50] similarity for $R_{\lambda}\geq 8$. A more detailed study of
this feature (Refs. [51, 52]) has shown that this universality is at best
approximate, but may still be regarded as a first approximation.

##  5 Results from further considerations

###  5.1 The decay exponents

We plot in figures 6 the decay exponents for the chosen cases with
$\kappa^{4}$ and $\kappa^{2}$ variation near the origin. The decay exponent is
close the theoretical value of $10/7$ predicted by the Kolmogorov theory for
the $\kappa^{4}$ case, and to the theoretical value of $6/5$ predicted by the
Birkhoff-Saffman theory for the $\kappa^{4}$ case. Cases 1.14 and 1.7 have a
convincing plateau for long periods of time although exponent for the case 1.6
seems to vary slowly with time slowly for the $\kappa^{4}$ case; similarly for
the $\kappa^{2}$ case, 2.14 and 2.7 seem to be constants even though 2.6
departs from its plateau behavior somewhat abruptly for $t/T_{E}>500$ or so.
The constants for the $\kappa^{4}$ and $\kappa^{2}$ cases are unmistakably
different.

(a) (b)

Figure 6:  The decay exponents for all three plausible cases of (a) $k^{4}$
and (b) $k^{2}$. The red dashed lines correspond to the theoretical exponent:
(a) $n=10/7$ and (b) $6/5$. The black dashed lines correspond to (a) $10/7\pm
2.5\%$ and (b) $6/5\pm 2.5\%$.

(a) (b)

Figure 7:  Evolution of the Taylor microscale Reynolds number as a function of
time for the best cases with (a) $k^{4}$ and (b) $k^{2}$ behaviour near the
origin.

(a) (b)

Figure 8:  Velocity derivative skewness for the decay cases examined. The
skewness is not far from the standard value of $-0.5$ expected for fully
turbulent turbulence.

Figure 7 shows that the Reynolds number remains moderate in all cases as the
decay proceeds. Cases 1.7 and 2.7 have the lowest Reynolds numbers, so we may
expect some deviations in other considerations below. Further, the skewness of
the longitudinal velocity derivative behaves approximately as expected, at
least for $t/t_{E}<500$ (see Fig. 8). An interesting feature of this figure is
that, although the Reynolds numbers initially range from 155 to 422, sustained
skewness of $-0.5$ is obtained only after an initial adjustment period at
which point $R_{\lambda}$ falls to about 40 and the skewness reaches its
maximum. This suggests that the decaying turbulence could have a long
transient.

Taken together, the basic result appears to be that the flow fields with
suitably controlled initial conditions evolve approximately into universal
decay states expected by the Kolmogorov and the Birkhoff-Saffman forms. It has
not been possible, despite considerable effort invested by us (see table 1 and
the discussion section), to obtain closer conformity to the theories.

###  5.2 Growth of integral length scales

The decay process eliminates small scales preferentially so that the integral
scales grow. It follows from (2.2) that the expected behaviors are

$L\sim t^{2/7}\hskip 56.9055ptL\sim t^{2/5}.$ (11)

for Kolmogorov and the Birkhoff-Saffman cases, respectively. We evaluate the
length scales using the formula stated below (1.3) and examine their
behaviours in Figs. 9 and 10, in log-log form and compensated forms,
respectively. One can improve the agreement in the log-log plot between the
theory (shown by the dashed line) and measurements by varying the virtual
origin, but we simply show the effect of using one with three initial integral
time scales. While the agreement with expectations is not perfect in either
case, it appears to be reasonable.

(a) (b)

Figure 9:  Scaling of integral length with time for (a) $\kappa^{4}$ and (b)
$\kappa^{2}$; $t_{vir}/T_{e}=3$, but other nearby choices for it do not make a
huge difference to our conclusions.

(a) (b)

Figure 10:  Normalized integral length scale for best cases with (a) $k^{4}$
and (b) $k^{2}$ spectra at the origin initially.

(a) (b)

Figure 11:  Evolution of the exponent $q$ of the energy spectrum of the form
$E\left(\kappa\right)\propto\kappa^{q}$ at low wavenumbers computed using just
the first three wavenumbers for the cases chosen. A constant with time
indicates permanence of large eddies. (a)
$E\left(\kappa\right)\propto\kappa^{4}$ and (b)
$E\left(\kappa\right)\propto\kappa^{2}$. Only for short periods of time is the
slope close to 4 in (a) and to 2 in (b). The erosion sets in relatively
quickly in all simulations of table 1.

##  6 Discussion and Conclusions

We have studied the $\kappa^{4}$ and the $\kappa^{2}$ cases using well-
resolved DNS for homogeneous and isotropic turbulence (HIT) for a variety of
conditions (table 1). Our results show that for an unambiguous power law for
energy decay to exist, one needs to start with high enough Reynolds number and
the energy-containing wavenumber is far enough away from dissipation scales,
while allowing a power law at the origin. Only a few cases we examined satisfy
these criteria. These cases also obey the requirement on the skewness of the
velocity gradient. We found that when the power-law at the origin was
initiated with a $\kappa^{4}$-type behavior, the Kolmogorov decay law with the
exponent close to $10/7$ was indeed observed. The $\kappa^{2}$ turbulence
conforms just as well to its own expectations.

At the end of this exercise, it is somewhat disappointing that the results are
not more closely aligned with theoretical arguments. We believe that this is
due to the fact that the $\kappa^{4}$ and $\kappa^{2}$ behaviors at the origin
do not persist for long periods of time; at least for the conditions of the
present simulations, nonlinear interactions diminish their extent rather
rapidly; see Fig. 11 which plots the slope neat the origin as a function of
time for the cases examined. This suggests that in neither case do the
invariants exist exactly except approximately for an initial period of time.
Indeed, as argued by Yakhot [53], that approximate condition seems to be
adequate for the Kolmogorov decay law; one presumes likewise for the
$\kappa^{2}$ behavior. Overall, it appears that for a power-law decay in
energy to exist according to one of the theories, one should set off the
initial conditions correctly even if one may not require strict permanence of
the large scale through out the decay. This appears true for exponents for the
energy decay and the length scale growth.

While considering the decay problem recently, Mccomb [54] made a number of
interesting theoretical points. One of them is that the Birkhoff-Saffman
invariant is strictly zero for HIT. He also pointed out that this does not
rule out that some such entity may exist for computational turbulence in a
finite box, which is essentially what we have found. The somewhat
unsatisfactory results we have found show that it is hard to satisfy all the
conditions exactly, even in box-type HIT, but our demonstrations suggest that
one can approach them reasonably closely. It is not clear to us, after our
experience with the many simulations listed in table 1, that permanence of
large scales can be established in any finite system.

Finally, it is worth commenting on the diversity of exponents observed in
experiments and simulations (Fig. 1). Indeed, if one did not pay any attention
to maintaining a particular spectral behavior near the origin, a range of
decay exponents were observed in simulations (often there were no good power
laws). However, some correspondence with the theoretical expectations were
obtained when we started the spectrum at the origin in a certain way. There is
experimental evidence to suggest that initial conditions set off the decay
with certain characteristics [38]. We speculate that the experimental
conditions are influenced by several of these issues. Besides, the practice of
choosing arbitrary virtual origins to fit the limited range of decay data by
power-laws, and of choosing to fit distances as close to the grids of 5 mesh
lengths, could result in a plethora of uncertain results. We think that Fig. 1
reflects these uncertainties.

Acknowledgments: We are pleased to present this short paper in the collection
honoring the remarkably long and productive scientific career of Professor
Uriel Frisch. We acknowledge support from the National Science Foundation
(Grant No. 1605914) and from the Extreme Science and Engineering Discovery
Environment (XSEDE) for computational resources.

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