**Appendix 2.F Supplementary Tables**

**3.3 Results and Discussion**

**3.3.5 Estimating the transient energy dissipation rate**

10^{−2} 10^{−1} 10^{0} 10^{1} 10^{2} 10^{3} 10^{4}
*R/L*11

10^{−2}
10^{−1}
10^{0}
10^{1}
10^{2}
10^{3}

*δ**G*(*R*)

50%

0 1000 2000 3000 4000 5000 6000

*R**λ*

10^{2} 10^{3} 10^{4}

*R**λ*
10^{2}

10^{3}
10^{4}
10^{5}
10^{6}
10^{7}

*R*0*.*5*/η**K*

Eq. (42)

∝R^{α}*λ*
DNS
VDTT

### A

10^{−2} 10^{−1} 10^{0} 10^{1} 10^{2} 10^{3} 10^{4}
*R/L*11

10^{−2}
10^{−1}
10^{0}
10^{1}
10^{2}
10^{3}

*δ**I*2(*R*)

50%

0 1000 2000 3000 4000 5000 6000

*R**λ*

10^{1} 10^{2} 10^{3} 10^{4}

*R**λ*
10^{2}

10^{3}
10^{4}
10^{5}
10^{6}
10^{7}

*R*0*.*5*/η**K*

Eq. (41)

∝R^{α}*λ*
VDTT
DNS

### B

**Figure 3.7** Random errors*δ*_{G}(*R*) (A) and*δ*_{I2}(*R*) (B) as a function of re-scaled averaging
window size *R/L*11 obtained from VDTT data at various R*λ* shown by the colorbar. The
analytical results for *δ*_{G}(*R*) (A, Eq. (3.46)) and *δ*_{I2}(*R*) (B, Eq. (3.45)) are shown by the
dashed black lines. The dotted black line annotated with “50%” in each subplot corresponds
to 50% error-threshold. The insets show the sizes of the averaging windows in terms of
*η*_{K} when *δ*_{G,I2}(*R*) ≤ 0*.*5 as a function of Taylor microscale Reynolds number R*λ*. The
inset plots include data from both DNS (red triangles) and the VDTT (grey circles). DNS
data used for the inset plots are from cases 1.3, 2.3 and 3.3 with *I* = 10% and *θ* = 0°.

The solid, blue lines show the prediction of the required averaging window according to
Eq. (3.50) (A-inset) and Eq. (3.49) (B-inset). The black dash-dotted line in inset plots is
a fit to the data: log*R/η**K* = ^{3}_{4}log_{20}^{3} −2 log*a*fit+*α*log R*λ* yielding *α* = 1*.*70±0*.*18 and
*a*_{fit}= 1*.*67±0*.*64 (A-inset); log*R/η*_{K} = log^{9}_{4}_{20}^{3} ^{3/4}−2 log*a*_{fit}+*α*log R*λ*yielding*α*= 1*.*57±0*.*09
and *a*_{fit} = 0*.*95±0*.*32. (B-inset)

deviation at high R*λ* can be explained, at least in part, by the strong assumptions made
for the derivation of the random errors, i.e., the equations (3.45), (3.43), and (3.46). In
particular, for experiments with high Re in VDTT, the assumption of Gaussian velocity
fluctuations with zero skewness is questionable, as shown in Fig. 3.15. Lenschow, Mann,
and Kristensen [56] has already established that the size of the averaging window for a
skewed Gaussian process [see Eq. (19) in 56] must be twice as large as for a Gaussian
process with vanishing skewness. However, further work is needed to investigate these
deviations and improve the theoretical prediction.

dissipation rate follow the ground-truth reference ⟨*ϵ*0(* x, t*)⟩

*R*or not. Respecting the intermittent nature of turbulence and energy dissipation, the standard deviation of

⟨*ϵ*0(* x, t*)⟩

^{R}is a first proxy for the variability of the trend in⟨

*ϵ*0(

*)⟩*

**x**, t^{R}. Hence, detecting the true trend requires that

*β*

*i*and

*δ*

*i*(

*R*) are smaller than the standard deviation of

⟨*ϵ*0(* x, t*)⟩

^{R}.

### 0*.*0 0*.*5 1*.*0 1*.*5 2*.*0

*/* ˙ *E*

h0(* x, t*)i

*R*h

*G*i

*R*h

*I*2i

*R*h

*L*i

*R*

### 0 13100 26300 39500 52700 65900 *r/η*

_{K}

### 0*.*0 0*.*5 1*.*0 1*.*5 2*.*0

*/* ˙ *E*

h_{0}(* x, t*)i

*R*h

_{I2}i

*R*h

_{I2}i

*R*(1±

*δ*˜

_{I2}(

*R*))

**Figure 3.8**(Upper plot) Proof of concept for estimating the coarse-grained energy dissipation
rate ⟨*ϵ*_{0}(* x, t*)⟩

*R*re-scaled by the energy injection rate ˙

*E*via the one-dimensional surrogates

⟨*ϵ**G*⟩*R*, ⟨*ϵ**I*2⟩*R*, and ⟨*ϵ**L*⟩*R* for R*λ* = 142, *R/η**K* ≈ 5500, *θ* = 0° and a turbulence intensity
*I* = 10% (DNS 2.0). All estimates are re-scaled by the energy injection rate ˙*E*, too. We
narrowed the fit-range to 20*η**K* ≤ *r* ≤ 200*η**K* ensuring optimal fit results. (Lower plot)
Comparison between ⟨*ϵ**I*2⟩*R**/E*˙ with estimated random error according to Eq. (3.45) for the
averaging window *R* and ⟨*ϵ*_{0}(* x, t*)⟩

*R*.

It can be already concluded from Figs. 3.2, 3.7, 3.10 and 3.14 that *ϵ**G* is the most
promising candidate to capture the true trend. However, to fully answer the above
questions, we need to conduct more in-depth analysis. The upper plot in Fig. 3.8
shows the re-scaled and coarse-grained dissipation field ⟨*ϵ*0(* x, t*)⟩

*R*for a sliding window of size

*R*≈ 5500

*η*

*K*and a turbulence intensity

*I*= 10% obtained from track of one virtual probe for case DNS 2.0 (“probe 0”). Consistent with results shown earlier,

⟨*ϵ**G*⟩*R* follows ⟨*ϵ*0(* x, t*)⟩

*R*best in comparison with ⟨

*ϵ*

*I*2⟩

*R*and ⟨

*ϵ*

*L*⟩

*R*. Both ⟨

*ϵ*

*I*2⟩

*R*and

⟨*ϵ**L*⟩^{R} are associated with substantial scatter, although ⟨*ϵ**I*2⟩^{R} has smaller deviations
from the ground-truth overall. Other probe tracks sample different portions of the
flow which is why a quantitative conclusion is not possible from one single probe. A
more comprehensive evaluation of which method is able to capture the true trend is
conducted below.

The lower plot in Fig. 3.8 shows ⟨*ϵ**I*2⟩^{R} together with the random error of *ϵ**I*2

as defined by Eq. (3.45). Despite the strong scatter, the ground-truth reference is
nearly always within the errorbar of *ϵ**I*2 with some exceptions, e.g. *r/η**K* *<* 5000 or
*r/η**K* ≈ 44000. It can also be seen that ⟨*ϵ**I*2⟩^{R} is, if at all, only weakly correlated
with the ground-truth reference ⟨*ϵ*0(* x, t*)⟩

^{R}for a window size of

*R/η*

*K*≈5500. This shows that it is extremely difficult, if at all possible, to track the true trend with low-resolution time records, which prevents the use of the gradient method.

0 1000 2000 3000 4000 5000 6000 7000
R*λ*

0*.*0
0*.*2
0*.*4
0*.*6
0*.*8
1*.*0

PearsonCorrelationCoeff.

*I*230*L*11

*L*30*L*_{11}

0 5000 10000 15000 20000

*R/η**K*

0*.*0
0*.*2
0*.*4
0*.*6
0*.*8
1*.*0

PearsonCorrelationCoeff. ^{}^{G}^{}^{R} ^{}^{I2}^{}^{R} ^{}^{L}^{}^{R}

### A B

**Figure 3.9**A: Dependence of the Pearson correlation coefficient between⟨*ϵ**i*⟩*R*and⟨*ϵ*0(* x, t*)⟩

*R*

as a function of the re-scaled averaging window*R/η*_{K} where *i*∈ {*G, I*2*, L*}. Time records of
the longitudinal velocity by all virtual probes and⟨*ϵ*_{0}(* x, t*)⟩

*R*are taken from DNS 2.0 with R

*λ*= 142, turbulence intensity

*I*= 10% and perfect alignment (

*θ*= 0°). The shaded region is given by the standard error. B: Dependence of the Pearson correlation coefficient between

⟨*ϵ*_{I2,L}⟩*R* and ⟨*ϵ*_{G}⟩*R*as a function R*λ* for a fixed re-scaled averaging window *R*= 30*L*_{11}. The
error bars of the ensemble averaged coefficients are given by the standard error.

To assess this correlation more quantitatively, we evaluate Pearson’s correlation
coefficient between the ground-truth reference ⟨*ϵ*0(* x, t*)⟩

^{R}and

*ϵ*

*G*,

*ϵ*

*I*2 as well as

*ϵ*

*L*, respectively, as a function of the re-scaled averaging window size

*R/η*

*K*for all virtual probes of case DNS 2.0. As an example, Pearsons correlation coefficient between

*ϵ*0(

*)⟩*

**x**, t^{R}and

*ϵ*

*I*2 is 0.33 in Fig. 3.8 (upper plot). Figure 3.9A shows the ensemble averages of Pearson’s correlation coefficient together with the standard error (shaded

area). While ⟨*ϵ**G*⟩*R* has a pronounced correlation with the ground-truth reference

⟨*ϵ*0(* x, t*)⟩

^{R}, both ⟨

*ϵ*

*I*2⟩

^{R}and ⟨

*ϵ*

*L*⟩

^{R}are only very weakly correlated with ⟨

*ϵ*

*G*⟩

^{R}.

The effect of R*λ* on Pearson’s Correlation coefficient is shown in Fig. 3.9B also
for the VDTT experiments at various R*λ*. Here, we compare *ϵ**I*2 and*ϵ**L* to *ϵ**G* in the
absence of ground-truth. To ensure a negligible systematic error, we chose a fixed
averaging window of *R*= 30*L*11 for each R*λ*. Figure 3.9B shows that the correlation for
*ϵ**I*2 is always higher than that of*ϵ**L* except for very low R*λ*. There is a non-monotonic
behavior in the correlation coefficients in Fig. 3.9B that seems to be related to the
skewness values shown in Fig. 3.15. Nonetheless, there is a clear increase in correlation
coefficients with R*λ*. Firstly, the random error of *δ**I*2(*R*) ranges from 20% to 40% at
*R* = 30*L*11. Secondly, the kurtosis of the instantaneous energy dissipation field scales
with R^{3/2}_{λ} [5, 49] which is why the variability in the instantaneous energy dissipation
field increases with R*λ*. Hence, at small R^{3/2}_{λ} and *R* = 30*L*11, ⟨*ϵ**I*2⟩^{30L}11 scatters only
randomly around the global mean energy dissipation rate (with a 3% standard deviation
of ⟨*ϵ**G*⟩30*L*11), which is why the correlation coefficient is low. In contrast, at large R*λ*

and *R* = 30*L*11, the locally averaged mean energy dissipation rate ⟨*ϵ**G*⟩^{30L}11 fluctuates
stronger (≈30% standard deviation of ⟨*ϵ**G*⟩^{30L}11) where *δ**I*2(*R*) is already comparable.