**Appendix 2.F Supplementary Tables**

**3.3 Results and Discussion**

**3.3.1 Verification of the analytical methods and a first insight into**

To verify the implementation of our methods, only data from cases with a low turbulence
intensity of 0.01 and an averaging window covering the entire size of the probe track are
used in this section. Furthermore,*ϵ**I*2and*ϵ**I*3are obtained by a fit according to Eq. (3.15)
with *n* = 2 and *n* = 3, respectively, in the inertial range with *r* ∈ [20*η**K**,*500*η**K*] for
DNS 2.1 and 3.1. Analogously, *ϵ**S* is inferred from the inertial range fit, Eq. (3.17),
in the range *f* ∈[*U/*(500*η**K*)*, U/*(20*η**K*)]. Due to the absence of an inertial range for
low Taylor-scale Reynolds number (DNS 1.1 with R*λ* = 74, compare Fig. 3.16), the
maximum of Eq. (3.15) is used to infer *ϵ**I*2 and *ϵ**I*3.

The distribution of the mean energy dissipation rate estimated by*ϵ**G**, ϵ**I*2*, ϵ**I*3*, ϵ**S*, and
*ϵ**L* for each probe at R*λ* = 302 is shown in Fig. 3.2. Estimations for other R*λ* are shown
in supplementary Fig. 3.10. The ground-truth reference for the mean energy dissipation

### _{ref} _{G} _{I} _{3} _{I} _{2} _{S} _{L} 0

### 2 4 6 8 10

*/* ˙ *E*

### h _{0} (**x**, t) i ^{N} h _{i} i ^{N}

**x**, t

**Figure 3.2** Validation of estimating the energy dissipation rate from *ϵ**G**, ϵ**I*2*, ϵ**I*3*, ϵ**S*, and *ϵ**L*

re-scaled by the energy injection rate ˙*E*. The data are taken from DNS 3.1 with 1000 probes,
R*λ* = 302, *I* = 1%, *θ* = 0° and maximal available averaging window (*R* ≈ 3550*η*_{K}). The
ensemble mean of each method⟨*ϵ**i*⟩*N* is denoted by red dots where the whiskers extend from
the minimal to maximal estimate of*ϵ*_{i} where*i*∈ {*G, I*3*, I*2*, S, L*}. The reference mean energy
dissipation rate for each probe is given by*ϵ*_{ref}. The dashed line represents the re-scaled global
mean energy dissipation rate of DNS 3.1 which is approximated by the ensemble average of
the true mean energy dissipation rate along the trajectory of each virtual probe.

rate for each probe is given by*ϵ*ref =⟨*ϵ*0(* x, t*)⟩ along the probe trajectory

*, which is the average of the instantaneous energy dissipation rate along the trajectory of each individual virtual probe (mean, median, standard deviation, range of*

**x***β*ref: 0%, −0

*.*7%, 18

*.*6%, −50%

*. . .*68

*.*2%) where

*β*ref =⟨

*ϵ*0(

*)⟩*

**x**, t*/*⟨⟨

*ϵ*0(

*)⟩⟩*

**x**, t*N*−1. The best performing method is the gradient method

*ϵ*

*G*(mean, median, standard deviation, range of

*β*

*G*:

−0*.*5%*,*1*.*7%, 19*.*3%, −48*.*1%*. . .*75*.*4%). The range of *β**G*, i.e. −49*.*8%*. . .*68*.*2%., is
also very close to the range of *β*ref. The method with highest error is *ϵ**I*3 (mean,
median, standard deviation, range of *β**I*3: 49*.*2%, 10*.*1%, 59*.*6%,−93*.*1%*. . .*822*.*2%).

The superior performance of *ϵ**G* compared to others is mainly due to the fact that it
relies on second-order dissipative statistics that can be captured with fast statistical
convergence within a short sampling interval. Hence, the distribution of*ϵ**G* and*ϵ*ref are
similar. *ϵ**I*3, on the other hand, relies on third-order moments of the velocity increments
of inertial scales associated with slower statistical convergence compared to*ϵ**G*. Hence,
longer velocity records under stationary conditions are needed. For this reason, the
third-order structure function is not considered further in this study, as one of the

main objectives of this study is to evaluate different methods suitable for extracting the time-dependent energy dissipation rate.

Fig. 3.2 also shows that the estimates of the energy dissipation rate provided by
*D**LL*(*r*) and *E*11(*κ*1) are close to each other, which can be explained by the fact that
they are both second-order quantities (in real and Fourier space, respectively) connected
by *f*(*r*). Furthermore, *ϵ**I*2 tends to overestimate the energy dissipation rate as the
mean is 8.8% higher than ⟨*ϵ*(* x, t*)⟩

*N*(median

*β*

*I*2 ∼3

*.*1%, standard deviation 41

*.*7%,

−69*.*6% *< β**I*2 *<* 199*.*8%). *ϵ**S* exhibits a similar overestimation (mean *β**S* ∼ 4*.*1%,
median *β**S* ∼ −2*.*0%, standard deviation 32*.*1%, −55*.*2%*< β**S* *<*210*.*4%), though to
a lesser extent. However, *ϵ**S* depends much stronger on properly setting the fit-range
than *ϵ**I*2 (supplementary Fig. 3.11). The spectral method *ϵ**S* can differ by a factor of
2 from *ϵ**I*2 depending on the high-frequency limit. This factor of 2 is in accordance
with a comparison of*ϵ**I*2 and *ϵ**S* by a linear fit resulting in a slope close to 0.5 [154].

In the DNS, the power spectrum is subject to strong statistical uncertainty at high
frequencies without ensemble-averaging the spectra of each virtual probe or longer
DNS runtimes. As the high-frequency limit of the inertial range of the spectrum is
hardly distinguishable from its dissipation range, the choice of the fit-range range for
*ϵ**S* is related to the fit-range of the longitudinal second-order structure function by
*f* ∈[*U/*(500*η**K*)*, U/*(20*η**K*)] as mentioned above. Wacławczyk et al. [153] found that the
estimation of the energy dissipation rate from the power spectral density is generally
robust at small wavenumbers whereas the second-order structure function performs
better at larger wavenumbers. With our choice of the fit-range *r* ∈ [20*η**K**,*500*η**K*],
we confirm that *ϵ**I*2 is already reliable at the lower end of the inertial range where
dissipative effects are negligible.

At last, *ϵ**L* overestimates ⟨*ϵ*(* x, t*)⟩

^{N}by 40% on average (median

*β*

*L*∼ 31

*.*5%,

−69*.*5%*< β**L**<*352*.*9%). This systematic overestimation might be due to the difficulty
in determining *L*11 as different methods for estimating the integral length *L*11 can
contribute to the systematic bias of *ϵ**L*. As mentioned above, we infer the longitudinal
integral length from fitting *f*(*r*) to the first zero crossing which yields, at least in the
DNS of this work, a systematic underestimation, as illustrated in Fig. 3.12. Figure 3.12
and 3.13 suggest that the scatter of *ϵ**L* is affected by the scatter of both *σ**u*^{′}_{1} and
*L*11. However, the accuracy of the dissipation constant *C**ϵ*, which is a function of
large-scale forcing and initial conditions [47, 145, 172, 173], can potentially cause larger
mean deviations of *ϵ**L* from ⟨*ϵ*(* x, t*)⟩

^{N}. Advantageously, the large-scale estimate

*ϵ*

*L*is applicable to low-resolution measurements and only weakly biased with respect to the ground-truth of DNS 1.1 where the variance is better converged (see Table 3.4 and Fig. 3.10).

To compare to the results obtained from DNS 3.1, the bias of each method for DNS
1.1 and 2.1 is presented in Table 3.4 (more details can also be found in the supplementary
Fig. 3.10). Both DNS 1.1 and 2.1 with R*λ* = 74 and R*λ* = 210, respectively, show
that *ϵ**G* reproduces the global reference ⟨*ϵ*(* x, t*)⟩

*N*closely (⟨

*β*

*G*⟩

*N*

*<*1%) as in DNS 3.1. In contrast,

*ϵ*

*I*3 is associated with the largest overestimates ⟨

*ϵ*(

*)⟩*

**x**, t^{N}for any R

*λ*. Both

*ϵ*

*I*2 and

*ϵ*

*S*show comparably small deviations from ⟨

*ϵ*(

*)⟩*

**x**, t^{N}. The mean

DNS ⟨*β**G*⟩^{N} ⟨*β**I*3⟩^{N} ⟨*β**I*2⟩^{N} ⟨*β**S*⟩^{N} ⟨*β**L*⟩^{N}
1.1 −0*.*003±0*.*001 0*.*132±0*.*005 −0*.*047±0*.*002 0*.*011±0*.*002 −0*.*044±0*.*003
2.1 −0*.*002±0*.*006 0*.*506±0*.*038 −0*.*011±0*.*014 0*.*074±0*.*010 0*.*313±0*.*017
3.1 −0*.*005±0*.*006 0*.*492±0*.*039 0*.*088±0*.*013 0*.*041±0*.*010 0*.*400±0*.*020
**Table 3.4** The systematic error of each method *β*_{i} relative to the global mean energy
dissipation rate, Eq. (3.21), of each DNS where *i*∈ {*G, I*3*, I*2*, S, L*}. The error is given by
the standard error which is defined as the standard deviation divided by the square root
of the number of samples. In both DNS 2.1 and 3.1, *ϵ*_{I2} and *ϵ*_{I3} were obtained by fitting
Eq. (3.15) for *n*= 2 and*n*= 3, respectively, in the range*r*∈[20*η**K**,*500*η**K*]. This fit-range is
also used for calculating*ϵ*_{S} and it was converted into frequency domain by*f* =*U/r*, where*U*
is the mean velocity. In the case of DNS 1.1, the maximum of Eq. (3.15) was used to infer*ϵ*_{I2}
due to the absence of a pronounced inertial range. We used the maximum available window
size*R* in all cases, fixed turbulence intensity *I* = 1% and considered perfect alignment, i.e.

*θ*= 0°.

relative deviation of *ϵ**S* due to the change in fit-range is much larger than the mean
values of Table 3.4. This is why the fit-range was chosen to be constant for DNS 2.1
(*r*∈[20*η**K**,*500*η**K*] and*f* ∈[*U/*(500*η**K*)*, U/*(20*η**K*)]) like in DNS 3.1 in order to prevent
for additional influences due to varying fit-ranges. Within these boundaries of the
fit-range for ⟨*D**LL*(*r*)⟩^{N}, the scaling exponent varied by approx. ±30%. It is found
that the systematic errors of *ϵ**I*2 and *ϵ**S* are smaller and comparable to those of the
gradient method when a fitting range is used for each dataset in which the scaling of the
structure function is close to the expected scaling, e.g. fit-range of*r* ∈[50*η**K**,*100*η**K*]
for DNS 2.1.

While *ϵ**L* is in close agreement with⟨*ϵ*(* x, t*)⟩

^{N}for DNS 1.1 (R

*λ*= 74), it strongly deviates for DNS 2.1 and 3.1, as elaborated above. In this context, we suggest that large deviations for

*ϵ*

*L*and

*ϵ*

*I*3are due to statistical uncertainties as DNS 2.1 and 3.1 are shorter in terms of the integral length scale

*L*11 than DNS 1.1 (Table 3.2, Fig. and 3.12).

However, although the box size of all DNS (see Table 3.2) is on the order of 6 integral
length scales, the ensemble of all probes represent a velocity record with a total length
of O(10^{5}) integral length scales for DNS 1.1 so that the statistics are well converged
(see Fig. 3.12A and 3.12B, for instance). Considering the reasonable agreements we
found between the mean energy dissipation rate estimated by different methods for
DNS 1.1 (Table 3.4) and the global value, we can verify the correct implementation
of the different methods studied here within the limits of the different methods and
statistical convergence.