The turbulent kinetic energy dissipation rate is one of the most important quantities characterizing turbulence. Experimental studies of a turbulent flow in terms of the energy dissipation rate often rely on one-dimensional measurements of the flow velocity fluctuations in time. In this work, we first use direct numerical simulation of stationary homogeneous isotropic turbulence at Taylor-scale Reynolds numbers

Turbulence is fundamental to many natural and engineering processes, such as transport of heat and moisture in the Earth's atmosphere (e.g.,

The instantaneous energy dissipation field

Apart from the instantaneous dissipation field

For a statistically stationary homogeneous isotropic (SHI) turbulent flow,

Then the global mean energy dissipation rate

In the case of atmospheric flows, in situ measurements made via airborne (e.g.,

Our literature review indicates that a systematic investigation is still needed to fully understand how the choice of averaging window, analysis methods, turbulence intensity and large-scale random flow velocities can influence estimating the mean energy dissipation rate and its deviations from the instantaneous energy dissipation rate. To this end, we systematically benchmark different techniques available in the literature using fully resolved DNS of statistically stationary, homogeneous, isotropic turbulence. Since the full dissipation field is available from DNS, this approach provides a ground-truth reference for comparisons with various estimation techniques. To bridge the gap between typical

Suppose

Most methods used to retrieve the dissipation rate require spatially resolved velocity statistics, although the velocity is recorded only at a single point and as a function of time in many experiments. Therefore, prior to estimating the energy dissipation rate, the one-dimensional velocity time record should first be mapped onto a spatially resolved velocity field. This is achieved by invoking Taylor's hypothesis

Generally, we have to distinguish between the global mean velocity

The mean of a one-dimensional velocity time record in the longitudinal direction

In certain circumstances, it is possible to map

One way to cope with non-stationary velocity time records is to evaluate the mean velocity for a subset of this signal. If the averaging time

The energy dissipation rate can be derived from various statistical quantities. A non-exhaustive list of the most common methods applicable to single-point measurements is shown in Table

Various definitions of the energy dissipation rate from the dissipative and inertial sub-range to the energy injection range. Here, the definitions for various dissipation estimates are given in the space or wavenumber domain, where

Proceeding from the Navier–Stokes equations for an incompressible Newtonian fluid, the instantaneous energy dissipation rate is given by

Averaged over a sphere with radius

Kolmogorov's second similarity hypothesis from 1941

In the inertial range, the transverse second-order structure function

According to K41

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.,

Usually, the longitudinal integral length scale

In this study, the direct numerical simulations of statistically homogeneous isotropic turbulent flow with

To mimic an ensemble of single-point measurements, we introduced 1000 virtual probes into the flow (one-way coupled, i.e., without back reaction on the flow), which move with a given constant speed in randomly directed straight paths to record the local flow velocity. Since the trajectories of the virtual probes are randomly oriented and the probability that they are exactly aligned with the simulation boundaries is low, the effect of periodic boundaries on the recorded velocity signal is expected to be small. We assume that the virtual probe records idealized velocity time series, neglecting the effect of transfer function associated with the anemometer

Using Taylor's hypothesis, the longitudinal velocity time series correspond to at least

Parameter overview for each DNS.

Schematic representation of the procedure for calculating energy dissipation rates from single-point velocity time records.

To evaluate the performance of different methods at Reynolds numbers applicable to atmospheric flows, we use the high-resolution hot-wire measurements of the longitudinal velocity components in the MPIDS VDTT (

The VDTT is a recirculating wind tunnel where the working gas SF

Longitudinal velocity fluctuations are temporally recorded with 30 to 60

The virtual probes record one-dimensional time records of the DNS longitudinal velocity component, from which the mean energy dissipation rate can be estimated by various methods and compared with the energy dissipation rate obtained directly from the DNS dissipation field. Generally, there are two different errors when estimating the mean energy dissipation rate, namely the systematic errors and random errors. The latter is related to the estimation variance of the mean energy dissipation rate, i.e., the statistical scatter of the

Systematic errors are an inherent feature of the methods used for estimating the dissipation rate but are also affected by experimental limitations and imperfections such as averaging windows and finite turbulence intensity parameterized by

In addition, the systematic error can be evaluated by comparing the estimates of the energy dissipation rate obtained by a method with imperfect data against the estimates obtained by the same method with optimal data. We denote these types of errors with

Estimates of the mean energy dissipation rate are susceptible not only to systematic errors, but also to random errors due to statistical uncertainty. For the averaging window, errors given by Eq. (

Different types of errors investigated in this study and their definitions. Subscript

In the following, we first focus on the DNS data to calculate

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,

Validation of estimating the energy dissipation rate from

Statistics (mean, median, standard deviation (SD) and range) of the reference-compared systematic errors for different methods and case DNS 3.1 with

The distribution of the mean energy dissipation rate estimated by

Figure

Lastly,

For a large turbulence intensity the local speed and direction of the flow vary significantly in time and space, which hinders the applicability of Taylor's hypothesis. Here, we quantify the impact of random sweeping on the accuracy of determining the mean energy dissipation rate. Therefore, we set the mean speed of the virtual probes in each DNS so that the turbulence intensity, and consequently the random sweeping, is a control parameter.

Figure

Systematic error

The fraction of track samples that can lead to a deviation of larger than 100 % increases from 0 % to

The effects of random sweeping on the energy dissipation estimates.

To quantify the impact of random sweeping on estimates of

Now let us consider the two inertial-sub-range methods. Here, as one can see in Figs.

For the spectral method, Eq. (

Overall, random-sweeping effects explain why the gradient method is more sensitive to turbulence intensity than inertial-range methods. Here, random sweeping accurately captures the deviations of the second-order structure function method as a function of turbulence intensity, whereas it can only partially account for the observed deviations for the spectral method.

In this section, we assess the influence of probe misalignment with respect to the mean flow direction on estimating the energy dissipation rate at the energy injection scale, the inertial range and the dissipation range. Here, we assume the angle

Analogously, the second-order structure function tensor is affected by misalignment (cf. Appendix

Influence of misalignment between probe orientation and the mean flow direction

To compare the analytical expressions with DNS results, the sensing orientation of the virtual probes is rotated around the

In experiments where the sensor can be aligned to the mean wind direction within

Here, our goal is to investigate how the accuracy of estimating the global mean energy dissipation rate depends on the averaging window size by investigating the associated systematic and random errors individually. To do this, we select an averaging window of size

Before comparing estimates of the energy dissipation rate using different methods, let us first compare the locally averaged energy dissipation rate

The effect of the averaging window size

We can further explore the influence of the averaging window

We now analytically focus on random errors associated with

Both the second-order structure function in Eq. (

The variance estimates are also subject to statistical uncertainty, which is also known as the random error in variance estimation

Consequently, the estimation of the mean energy dissipation rate by the scaling argument in Eq. (

Similarly, the longitudinal second-order structure function is also affected by the estimation variance

Thus, the uncertainty in estimating the variance propagates to

Equations (

Furthermore, Eqs. (

For example, for the random errors of

Figure

Random errors

The scaling of

Furthermore, it is evident from Fig.

For both

However, VDTT experiments with

As has been shown previously in Figs.

It can already be concluded from Figs.

The lower plot in Fig.

To assess this correlation more quantitatively, we evaluate the Pearson correlation coefficient between the ground-truth reference

The effect of

Up to this point, we have presented the results largely as is so that one can interpret them with minimal bias. However, the amount of data and details given may make the use of the results in practice difficult. Therefore, we propose here practical guidelines for measuring the energy dissipation rate from one-dimensional velocity records in atmospheric flows.

The gradient method should be preferred over other methods for conditions where the turbulence intensity is low and where the probe could be perfectly aligned in the direction of the mean wind. In particular, the gradient method is more sensitive to turbulence intensity than inertial-range methods due to random-sweeping effects. Low values of turbulence intensity and ideal alignment of probes can be best controlled in ground-based measurements. Measurements aboard research aircraft traveling at a true speed of 100 m s

Other estimates based on inertial-sub-range methods (cf. Table

Given atmospheric turbulence with

Our analysis further shows that estimating the transient energy dissipation in atmospheric clouds (

The large random errors also preclude the possibility of interpreting trends in the measured energy dissipation rate via the second-order structure function if we consider the results shown in Figs.

We have presented an extensive review on the analysis procedure for estimating the energy dissipation rate from single-point one-dimensional velocity time records along with an overview of the advantages and disadvantages (see Table

The main methods considered in this study are the gradient method

Each method could reproduce the ground-truth reference

In the case of finite turbulence intensities,

Considering the probe orientation, the gradient method (

We provide scaling arguments

The random error in the gradient method

Only

As discussed in detail below, the mean energy dissipation rate can be related to second-order statistics of the velocity field, either in terms of velocity gradients or in terms of velocity increments. In any case, the two-point velocity covariance tensor turns out to be the central quantity of interest from which the second-order structure function tensor, the spectral energy tensor and the velocity gradient covariance tensor can be obtained.

In the following, we assume a zero mean SHI turbulence so that two-point quantities depend only on the separation vector

Under the given assumptions, the two-point velocity covariance tensor takes the form (e.g.,

Analogously, a covariance tensor can be defined for velocity increments, i.e., the second-order velocity structure function tensor

The longitudinal second-order structure function

Furthermore, the velocity gradient covariance tensor can also be defined in terms of the velocity covariance tensor

The two-point velocity covariance tensor can be expressed in Fourier space through the energy spectrum tensor

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.

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

Nomenclature for the turbulent flow. If our naming convention differs from the terminology in

Continued.

Nomenclature for the subscripts.

The systematic error in each method

Validation of estimating the energy dissipation rate from

Estimates of the mean energy dissipation rate as a function of the fit range for

Convergence of higher-order statistical quantities and longitudinal integral length scales as well as small- and large-scale anisotropy obtained from all virtual probes of DNS 1.1 (

Different estimates of the integral length from DNS 3.1 with

Convergence of energy dissipation rate estimates for

Skewness and kurtosis of all VDTT experiments as a function of

Compensated longitudinal second-order structure functions for DNS 1.1

Resolution effect on

In the following, we illustrate how one obtains an expression for the impact of random-sweeping effects on the dissipation rate estimate in terms of the turbulence intensity using the gradient method

To numerically assess how random sweeping smears out the spectrum at finite turbulence intensities (see Fig.

Due to the misalignment, the probe frame of reference is rotated with respect to the frame given by the mean velocity. Without loss of generality, we assume that the misalignment is due to a rotation around the

The misalignment has two consequences. First, the estimated mean velocity differs from the true mean velocity, which leads to errors when evaluating Taylor's hypothesis. Second, the measured velocity component is not the true velocity component but rather a combination of the longitudinal and transverse components.

In the probe frame of reference, we assume that we measure the longitudinal velocity field component, i.e., the component along the probe orientation

The sampling direction of the probe, given by the direction of the mean velocity,

Combining these two aspects, the vectorial distance covered by the probe in terms of the estimated one, therefore, becomes

The same arguments that we applied to the covariance tensor, of course, also hold for the second-order structure function tensor, Eq. (

The misalignment error for the gradient method can be estimated analytically starting from the longitudinal component of the velocity gradient covariance tensor

The code can be shared by the corresponding author upon request.

The data can be shared by the corresponding author upon request.

TB, MW and EB provided the velocity data. MS, EB, MW and GB conceptualized the study. MS developed, validated and ran the analysis code. The theoretical modeling was performed by MS, TB and MW. MS, TB, MW and GB analyzed and interpreted the data. MS, TB, MW and GB wrote the initial draft. MS, TB, EB, MW and GB proofread and edited the final paper.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We thank Christian Küchler and Gregory Bewley for providing us with VDTT data, and we thank David Kleinhans, Christian Küchler, Freja Nordsiek, Naseem Ali and Holger Nobach for helpful discussions. Furthermore, we thank Cristian C. Lalescu and Bérenger Bramas for their support and development of the TurTLE code used in this study. We are grateful for the support by the MPIDS High-Performance Computation (HPC) team for providing and maintaining computational resources.

This work was supported by the Fraunhofer–Max Planck cooperation program through the TWISTER project. Marcel Schröder was financially supported by the Konrad-Adenauer-Stiftung. Tobias Bätge was financially supported by a fellowship of the IMPRS for Physics of Biological and Complex Systems. The article processing charges for this open-access publication were covered by the Max Planck Society.

This paper was edited by Luca Mortarini and reviewed by two anonymous referees.