**Appendix 2.F Supplementary Tables**

**3.2 Methods**

**3.2.3 Estimating the Energy Dissipation Rate**

of the velocity covariance tensor
*R**ijkl*(* r*) =

**∂u*^{′}_{i}(* x, t*)

*∂x**k*

*∂u*^{′}_{j}(* x, t*)

*∂x**l*

+

=−lim

*r*→0*∂**r**k**∂**r**l**R**ij*(* r*)

*.*(3.7) Since the local and instantaneous energy dissipation rate (cf. Eq. (3.11)) is defined in terms of the strain rate tensor

*S*

*ik*= (

*∂u*

^{′}

_{i}(

*)*

**x**, t*/∂x*

*k*+

*∂u*

^{′}

_{k}(

*)*

**x**, t*/∂x*

*i*)

*/*2, the mean energy dissipation rate can be directly related to contractions of the velocity gradient covariance tensor. Note that in a turbulent flow with zero-mean velocity, the strain rate tensor

*S*

*ik*equals the fluctuation strain rate tensor

*s*

*ik*.

The two-point velocity covariance tensor can be expressed in Fourier space through the spectral energy tensor [5]

Φ*ij*(* κ*) = 1
(2

*π*)

^{3}

Z Z Z +∞

−∞ *R**ij*(* r*)

*e*

^{−iκ·r}d

*(3.8)*

**r**,where* κ* is the wave vector. For SHI turbulence, Φ

*ij*(

*) takes the form Φ*

**κ***ij*(

*) =*

**κ***E*(

*κ*)

4*πκ*^{2}

*δ**ij* − *κ**i**κ**j*

*κ*^{2}

(3.9)

where*E*(*κ*) is the energy spectrum function.

Since access to the full energy spectrum function is not always available, one- dimensional spectra are of interest, too. The mean energy dissipation rate can be estimated from the inertial range scaling of the longitudinal one-dimensional spectrum (as shown in Eq. (3.17)), which can be calculated by both the energy spectrum function

and the velocity covariance tensor [5]

*E*11(*κ*1) = ^{Z} ^{∞}

*κ*1

*E*(*κ*)

*κ* 1− *κ*^{2}_{1}
*κ*^{2}

!

d*κ*= 1
*π*

Z ∞

−∞*R*11(* e*1

*r*1)

*e*

^{−iκ}

^{1}

^{r}

^{1}d

*r*1

*,*(3.10) with the wavenumber

*κ*1 corresponding to the

*1-direction and*

**e***R*11(0) = ⟨

*u*

^{′2}⟩ =

R∞

0 *E*11(*κ*1)d*κ*1.

This concludes the second-order statistics in terms of the velocity that we consider in the following to determine the mean energy dissipation rate.

rangeDissipationestimate(Eq.)symboldefinitionassumptiondissipativesub-rangeinstantaneous(3.11)*ϵ*02*ν*(*s**ij**s**ij*)(local)volumeaverage(3.12)⟨*ϵ*(* x,t*)⟩

*R*34

*π*

*R*3 RRR

_{V}

(*R*)*ϵ*0(* x*+

*)d*

**r**,t*SHI(longitudinal)gradient(3.14)*

**r***ϵ*

*G*15

*ν*

*∂*

*u*′1(

*)*

**x***∂**x*1 2 SHI2nd-orderSF(dissipationrange) ∗*ϵ**D*215*νD**LL*(*r*)*/r* 2SHI,*r*≲*η**K*zero-crossings ∗*ϵ*+15*π* 2*ν*⟨*u* ′21⟩*N* 2*L*SHIinertialsub-range4/5law ∗((3.15),*n*=3)*ϵ**I*3−5*/*4*D**LLL*(*r*)*/r*SHI,K412nd-orderSF(inertialrange)((3.15),*n*=2)*ϵ**I*2(*D**LL*(*r*)*/C*2) 3*/*2*/r*SHI,K41spectral(3.17)*ϵ**S* *κ* 5*/*31*E*11(*κ*1)
18*/*55*C**K* 3*/*2SHI,K41
cutofffilter ∗*ϵ**C* 23 2⟨*u*′2*C*⟩

18*/*55*C**K* _{κ} −2*/*31*,*low−*κ* −2*/*31*,*up 3*/*2SHI,K41
energyinjectionscalescalingargument(3.18)*ϵ**L**C**ϵ**σ* 3*u*′1 */L*11SHI(global)mean(3.13)⟨*ϵ*⟩lim*R*→∞ ⟨*ϵ*0(* x,t*)⟩

*R*SHI

**Table3.1**Variousdefinitionsoftheenergydissipationratefromthedissipativeandinertialsub-rangetotheenergyinjectionrange.Here,thedefinitionsforvariousdissipationestimatesaregiveninthespaceorwavenumberdomainwhere*ν*istheviscosity,*s**ij*isthevelocityfluctuationstrainratetensor,*R*istheradiusoftheaveragingvolumeV(*R*)(windowsizefor1Ddata),*u* ′1(* x*)isthelongitudinalvelocityfluctuationfieldalong

*x*1,

*D*

*L...*(

*r*)isthe

*n*th-orderlongitudinalstructurefunctionfordistance

*r*,⟨

*u*′21⟩isthevarianceof

*u*′1(

*),*

**x***σ*

*u*′1 isthestandarddeviationof

*u*′1(

*),*

**x***N*

*L*isthenumberofzerocrossingsofavelocityfluctuationsignalperunitlength,

*C*2≈2,

*E*11(

*κ*1)istheone-dimensionalenergyspectrumwithwavenumber

*κ*1,

*C*

*K*≈1

*.*5,

*u*′2

*C*isthevarianceofaband-passfilteredsignalforwavenumbers

*k*∈[

*k*low

*,k*up],

*C*

*ϵ*isthedissipationconstant,

*L*11isthelongitudinalintegralscale,and

*η*

*K*istheKolmogorovlengthscale.Dissipationestimatesindicatedwith ∗arenotconsideredindetailinthiswork.Theassumptionsofstationarity(S),homogeneity(H),localisotropy(I)andKolmogorov’ssecondsimilarityhypothesisfrom1941(K41)arerepresentedbytheirindividualabbreviations.Referencesaregiveninthecorrespondingsectionsinthemaintext.

**Dissipative sub-range**

Proceeding from the Navier-Stokes equations for an incompressible, Newtonian fluid, the instantaneous energy dissipation rate is given by [e.g. 5, 42]

*ϵ*0(* x, t*) = 2

*ν*(

*S*

*ij*

*S*

*ij*)

*.*(3.11) The contribution of the fluctuating part to the energy dissipation is much larger than the contribution of the mean flow in the case of high-Re turbulent flows [5, 138]. Hence, the instantaneous energy dissipation rate can be defined in terms of the velocity fluctuations only [5] where, in Eq. (3.11) and Table 3.1,

*S*

*ij*is replaced by the fluctuation strain rate tensor

*s*

*ij*= (

*∂u*

^{′}

_{i}(

*)*

**x**, t*/∂x*

*j*+

*∂u*

^{′}

_{j}(

*)*

**x**, t*/∂x*

*i*)

*/*2.

Averaged over a sphere with radius*R* and volumeV(*R*), the (local) volume average
of the instantaneous energy dissipation rate is [5]

⟨*ϵ*0(* x, t*)⟩

^{R}= 3 4

*πR*

^{3}

ZZ Z

V(*R*)*ϵ*0(* x*+

*)d*

**r**, t*. (3.12) The local volume average*

**r***ϵ*

*R*(

*) converges to the global mean energy dissipation rate if*

**x**, t*R*tends to infinity:

⟨*ϵ*⟩= lim

*R*→∞⟨*ϵ*0(* x, t*)⟩

^{R}=−

*ν*lim

|* r*|→0

*∂*

_{r}

^{2}

_{j}

*R*

*ii*(

*)*

**r**, t*,*(3.13)

where the right-hand-side follows from partial integration. In experiments, it is often
not possible to measure *ϵ*0(* x, t*). Under the assumption of statistically homogeneous
and isotropic turbulence, the volume/time averaged energy dissipation rate are typically
inferred from one-dimensional surrogates [136, 138, 140, 143, 163, 164, among others],
such as:

*ϵ**G*=−15*ν* lim

|* r*|→0

*∂*

^{2}

_{r}

_{1}

*R*11(

*) = 15*

**r**, t*ν*

* *∂u*^{′}_{1}(* x, t*)

*∂x*1

!2+

= 15*ν*
*U*^{2}

* *∂u*^{′}_{1}(*t*)

*∂t*

!2+

*,* (3.14)
where the mapping between space and time domains is possible by applying the Taylor’s
hypothesis if *σ*_{u}^{′}

1*/U* ≪1 [131, 140]. The relationship shown in Eq. (3.14) is often called
the “direct” method in the literature [131, 140, e.g.]. The deviation of *ϵ**G* from its
global mean ⟨*ϵ*⟩depends quadratically on the turbulence intensity [131, 137, 162, 163].

**Inertial sub-range: indirect estimate of energy dissipation rate**

Kolmogorov’s second similarity hypothesis from 1941 [43] provides another method for
estimating the energy dissipation rate in the inertial range. Based on the inertial range
scaling of the *n*th-order longitudinal structure function, the mean energy dissipation

rate can be calculated by [5]

*D**L...L*(*r*) = *C**n*(*ϵ**In**r*)^{ζ}^{n} ⇔*ϵ**In* = *D**L...L*(*r*)
*C**n*

!1*/ζ**n* 1

*r* *,* (3.15)

where *C**n* is a constant, e.g. *C*2 ≈2 [5], and *ζ**n*=*n/*3 according to K41 by dimensional
analysis. In practice, *ϵ**I*2 (Table 3.1) is retrieved either by fitting a constant to the
compensated longitudinal second-order structure function*D**LL*(*r*), *n* = 2 in Eq. (3.15),
or a power law (∝*r*^{2/3}) to the inertial range of*D**LL*, Eq. (3.6), if the inertial range is
pronounced over at least a decade. Accounting for intermittency, the scaling exponent
of the *n*th-order structure function is modified to *ζ**n* = ^{n}_{3}[1− ^{1}_{6}*µ*(*n*−3)] where *µ* is
the internal intermittency exponent [49]. The inertial range is bounded by the energy
injection scale *L* at large scales and by the dissipation range at small scales. That
is why the fit-range has to be chosen such that *η**K* ≪ *r* ≪ *L*. If the inertial range
is not sufficiently pronounced, the extended self similarity can be used to extend the
inertial range [165, 166]. Otherwise,*ϵ**I*2 can also be approximated by the maximum of
Eq. (3.15) (for *n*= 2) within the same range as before. This is possible because the
maximum lies on the plateau in case of a perfect K41 inertial range scaling.

In the inertial range, the transverse second-order structure function*D**N N*(*r*) is equal
to 4*D**LL*(*r*)*/*3 in a coordinate system where * r* =

*r*1 is parallel to the longitudinal flow direction [5] highlighting the importance of the measurement direction.

**e****Inertial sub-range: spectral method**

According to K41 [43], the inertial subrange of the energy spectrum function scales
as *E*(*κ*) ∝ ⟨*ϵ*⟩^{2/3}*κ*^{−5/3} with the wavenumber *κ* by dimensional analysis. In isotropic
turbulence, the energy spectrum function can be converted into a one-dimensional
energy spectrum*E*11(*κ*1) (Eq. (3.10)). The wavenumber space is not directly accessible
from time-resolved point-like velocity measurements. Given the validity of Taylor’s
hypothesis, the one-dimensional energy spectrum *E*11(*κ*1) transforms to the frequency
domain with*F*11(*f*) = 2*πE*11(*κ*1)*/U* where *κ*1 = 2*πf /U* [e.g. 137, 167] yielding:

*F*11(*f*) = 18*/*55*C**K*

*U*
2*πϵ**S*

2*/*3

*f*^{−5/3}*,* (3.16)

which yields

*ϵ**S* = 2*π*
*U*

*f*^{5/3}*F*11(*f*)
18*/*55*C**K*

!3*/*2

*,* (3.17)

with the one-dimensional Kolmogorov constant *C**K* = 1*.*5 [5, 168]. Applying Taylor’s
hypothesis to a flow with a randomly sweeping mean velocity causes the Kolmogorov
constant to be systematically overestimated whereas the scaling of power-law spectra
remains unaffected [158, 169]. Hence, Eq. (3.17) is still valid for a randomly sweeping

mean velocity although *ϵ**S* is overestimated if*C**K* is not corrected for random sweeping.

*F*11 has the units of a power spectral density m^{2}*/*s and ⟨*u*^{′2}_{1}⟩ = ^{R}_{0}^{∞}*F*11(*f*)d*f*.

Depending on the Fourier transform convention, the prefactor of *C**K* might vary [41,
e.g.]. Under the assumption of Kolmogorov scaling in the inertial sub-range, this
identity can be adopted to estimate the mean energy dissipation rate from low and
moderate resolution velocity measurements of a finite averaging window [87, 140, 142,
170].

**Energy injection scale**

In equilibrium turbulence, the rate at which turbulent kinetic energy is transported
across eddies of a given size is constant in the inertial range assuming high enough
Reynolds numbers [e.g. 171]. By dimensional argument, this rate is proportional to
*u*^{3}(*l*)*/l*, where*u*(*l*) is the characteristic velocity scale of eddies of length *l*. Considering
the integral scale *L*11 and its characteristic velocity scale *u*(*L*11), namely the RMS
velocity fluctuation*σ**u*^{′}_{1}, the mean energy dissipation rate can be calculated by [143]

*ϵ**L* =*C**ϵ*

*σ*_{u}^{3}^{′}

1

*L*11

*,* (3.18)

where *C**ϵ* is the dissipation constant and for time- and space-varying turbulence, it
depends on both initial as well as boundary conditions and the large-scale structure
of the flow [47, 145, 172, 173]. *C**ϵ* is found to be about 0.5 for shear turbulence [47,
174] and 1.0 [172, 175] (or 0.73 Sreenivasan [47]) for grid turbulence. In this work *C**ϵ*

is assumed to be 0.5 which holds approximately in a variety of flows [54, 168, and references therein].

Usually, the longitudinal integral length scale *L*11 is defined as [5]

*L*11= lim_{r→∞}^{Z} ^{r}

0 d*r*^{′}*f*(*r*^{′}) = *πE*11(0)

2⟨*u*^{′2}_{1}⟩ *,* (3.19)
However, due to experimental limitations,*r* is often given by the first zero-crossing
of *f*(*r*) in both laboratory and *in situ* measurements [54, e.g.], or, alternatively, by the
position where *f*(*r*) = 1*/e* [176, 177]. Griffin et al. [178] carried out an integration
for *r* → ∞ performing an exponential fit in the vicinity of *f*(*r*) = 1*/e*. Notably,
*E*11(0) =^{R}_{0}^{∞}d*κE*(*κ*)*/κ* so that the estimation of *L*11 from the power spectrum is only
recommended if *E*(*κ*) = ^{1}_{2}*κ*^{3 d}_{dκ}^{}^{1}_{κ}^{dE}_{dκ}^{11}^{(κ)}^{} [5] is accurately determined like in DNS.

This approach not only requires a fully resolved velocity measurement but also a well
converged *E*11(*κ*1) as the conversion is highly sensitive to statistical scatter.

Ultimately, the choice of*L*11 strongly affects *ϵ**L*. In this work, we integrate *f*(*r*) to
the first zero-crossing because it does not depend on assumptions on the decay of *f*(*r*)
and the choice of the fit-range.