the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Estimating the turbulent kinetic energy dissipation rate from onedimensional velocity measurements in time
Marcel Schröder
Tobias Bätge
Eberhard Bodenschatz
Michael Wilczek
Gholamhossein Bagheri
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 onedimensional measurements of the flow velocity fluctuations in time. In this work, we first use direct numerical simulation of stationary homogeneous isotropic turbulence at Taylorscale Reynolds numbers $\mathrm{74}\le {R}_{\mathit{\lambda}}\le \mathrm{321}$ to evaluate different methods for inferring the energy dissipation rate from onedimensional velocity time records. We systematically investigate the influence of the finite turbulence intensity and the misalignment between the mean flow direction and the measurement probe, and we derive analytical expressions for the errors associated with these parameters. We further investigate how statistical averaging for different time windows affects the results as a function of R_{λ}. The results are then combined with Max Planck Variable Density Turbulence Tunnel hotwire measurements at $\mathrm{147}\le {R}_{\mathit{\lambda}}\le \mathrm{5864}$ to investigate flow conditions similar to those in the atmospheric boundary layer. Finally, practical guidelines for estimating the energy dissipation rate from onedimensional atmospheric velocity records are given.
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Turbulence is fundamental to many natural and engineering processes, such as transport of heat and moisture in the Earth's atmosphere (e.g., Wyngaard, 1992; Garratt, 1994; Muschinski and Lenschow, 2001; Fairall and Larsen, 1986; Hsieh and Katul, 1997), wind energy conversion (Smalikho et al., 2013), entrainment and mixing (e.g., Warhaft, 2000; Sreenivasan, 2004; Deshpande et al., 2009; Gerber et al., 2008, 2013; Siebert et al., 2013; Fodor and Mellado, 2020), and warm rain initiation (e.g., Shaw, 2003; Devenish et al., 2012; Pumir and Wilkinson, 2016; Li et al., 2020), to name just a few. In threedimensional turbulence, the kinetic energy is typically injected into the flow at the largest scales and is successively transferred to smaller eddies by means of the direct energy cascade. At the smallest scales characterized by the Kolmogorov length scale (or the dissipation scale) η_{K}, kinetic energy is dissipated by viscous effects at the energy dissipation rate (a list of all the parameters and symbols used in this study can be found in Tables A1 and A2). The energy dissipation rate is one of the most fundamental quantities in turbulence and is used to estimate many relevant features of a turbulent flow, such as the Kolmogorov length scale η_{K}; the Taylor microscale λ; the Taylorscale Reynolds number R_{λ}; and, by means of dimensional estimates, the energy injection scale.
The instantaneous energy dissipation field ϵ_{0}(x,t), which is a function of the fluid kinematic viscosity ν and the velocity gradient tensor, is highly intermittent with strong smallscale fluctuations (Pope, 2000; Davidson, 2015, and references therein), which are at the core of the intermittency problem in turbulence (Sreenivasan and Antonia, 1997; Muschinski et al., 2004; Buaria et al., 2019). By “instantaneous” we want to emphasize here that ϵ_{0} is the energy dissipation rate at one point in space and time within the flow. It also plays an important role in turbulent mixing in reacting flows (e.g., Sreenivasan, 2004; Hamlington et al., 2012; Sreenivasan, 2019) or turbulenceinduced rain initiation in warm clouds (Devenish et al., 2012). ϵ_{0}(x,t), however, is extremely difficult to measure experimentally because it requires complete knowledge of the threedimensional velocity field with spatial and temporal resolution that can resolve scales smaller than or at least comparable to the Kolmogorov scales.
Apart from the instantaneous dissipation field ϵ_{0}(x,t), the energy dissipation in a turbulent flow can be statistically described by either the local or global mean energy dissipation rate, which are both important. Local volume averages of the instantaneous dissipation field 〈ϵ_{0}〉_{R} and related surrogates, e.g., diagonal or offdiagonal components of the velocity gradient, can capture intermittency of turbulence (Lefeuvre et al., 2014; Almalkie and de Bruyn Kops, 2012, and references therein). The local volume averages of the dissipation field converge to the global mean energy dissipation rate 〈ϵ〉 for statistically converged sampling. The mean dissipation rate 〈ϵ〉 can be used to parameterize the statistics of statistically homogeneous and locally isotropic turbulence based on Kolmogorov's phenomenology (K41) (Kolmogorov, 1941). Note that even if the global mean energy dissipation rate 〈ϵ〉 is known, it is also of high interest to know how locally averaged dissipation rates 〈ϵ_{0}〉_{R} differ from the global mean energy dissipation rate 〈ϵ〉. For example, the local dissipation rate determines whether droplets in a cloud behave as tracer or inertial particles, which in turn can affect the probability of collision and coalescence of the droplets and thus the likelihood of precipitation initiation (e.g., see Shaw, 2003).
For a statistically stationary homogeneous isotropic (SHI) turbulent flow, 〈ϵ〉 can be estimated from timedependent singlepoint onedimensional velocity measurements through different methods, such as longitudinal or transverse velocity gradients (Wyngaard and Clifford, 1977; Elsner and Elsner, 1996; Antonia, 2003; Siebert et al., 2006, among others), inertialrange scaling laws comprising the famous $\mathrm{4}/\mathrm{5}$ law (Kolmogorov, 1941, 1991; Muschinski et al., 2001), counting zero crossings of the velocity fluctuation time series (Sreenivasan et al., 1983; Wacławczyk et al., 2017), or dimensional arguments (e.g., Taylor, 1935; McComb et al., 2010; Vassilicos, 2015). These methods usually invoke Taylor's hypothesis to map temporal signals onto spatial signals, which requires a sufficiently small turbulence intensity. The turbulence intensity is defined as the ratio of the root mean square velocity fluctuations ${\mathit{\sigma}}_{{u}^{\prime}}$ to the mean velocity U. When all of these criteria are met, singlepoint velocity measurements with hotwire anemometers at a high temporal resolution have been shown to be suitable for accurately estimating the global energy dissipation rate (see also below) (Lewis et al., 2021; Sinhuber, 2015; Elsner and Elsner, 1996; Antonia, 2003; Frehlich et al., 2003). However, in atmospheric flows, the assumption of ideal stationary homogeneous isotropic turbulence needs to be considered very carefully, as, for example, thermals change in local weather conditions, and of course the diurnal cycle may lead to nonstationarity and inhomogeneity.
Then the global mean energy dissipation rate 〈ϵ〉 is not representative, as the turbulence can be highly time and spacedependent even at the energy injection scales. As a result, one needs to calculate a local 〈ϵ_{0}〉_{τ} and 〈ϵ_{0}〉_{R}, respectively, based on velocity statistics for a properly chosen averaging window τ in time and R in space, which is short enough for resolving the temporal or spatial variations but also long enough to obtain statistically representative values with acceptable systematic and/or random errors (e.g., Wyngaard, 1992; Lenschow et al., 1994). Therefore, a conflict arises with respect to the averaging time between resolving smallscale features of a turbulent flow and statistical convergence under nonstationary and inhomogeneous conditions.
In the case of atmospheric flows, in situ measurements made via airborne (e.g., Malinowski et al., 2013; Siebert et al., 2006, 2013; Muschinski et al., 2004; Frehlich et al., 2004; Nowak et al., 2021; Dodson and Small Griswold, 2021) as well as groundbased (e.g., Chamecki and Dias, 2004; O'Connor et al., 2010; Risius et al., 2015; Siebert et al., 2015) platforms can typically only resolve the coarsegrained time series of the local mean energy dissipation rate 〈ϵ_{0}〉_{τ}. However, it remains unclear how large the errors in estimating the coarsegrained time series of the local mean energy dissipation rate are due to individual choices of the averaging window, since there is currently no highresolution threedimensional velocity measurement available during in situ measurements to serve as the ground truth. In the absence of a groundtruth reference, the comparison between different methods was used as the next best benchmark for validity of a given method (Siebert et al., 2010; Wacławczyk et al., 2020; Siebert et al., 2006; Risius et al., 2015; Wacławczyk et al., 2017), which in some cases makes the interpretation of the data difficult due to the large discrepancies between the estimates obtained by different methods. As an example, Wacławczyk et al. (2020) found deviations of about 5 %–50 % with respect to estimating the mean energy dissipation rate depending on the method and averaging windows using synthetic data modeled via a von Kármán spectrum. Another example is the work of Akinlabi et al. (2019), who found that estimates of the mean energy dissipation rate by onedimensional longitudinal velocity can differ by a factor of 2 to 3 from those calculated using direct numerical simulations (DNSs), depending on the method used.
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 largescale 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 groundtruth reference for comparisons with various estimation techniques. To bridge the gap between typical R_{λ} of DNS and atmospheric flows, we use highresolution measurements of the longitudinal velocity components of the Variable Density Turbulence Tunnel (VDTT) (Bodenschatz et al., 2014; Sinhuber, 2015; Küchler et al., 2019) at various Taylorscale Reynolds numbers R_{λ} between 140 and 6000. The impact of turbulence intensity, largescale randomsweeping velocities, the size of the averaging window, the Reynolds number and also possible experimental imperfections such as anemometer misalignment are investigated in detail. Our work aims to be a step towards the goal of extracting the timedependent energy dissipation rate from nonideal naturally occurring turbulent flows, mitigating the impact of nonideal features of the flow, e.g., anisotropy or inhomogeneity. In Sect. 2, we first define the central statistical quantities and the individual methods for estimating the energy dissipation rate in detail. An analysis of the individual methods including discrepancies, errors due to finite turbulence intensity and alignment errors are discussed in Sect. 3 followed by a summary of our findings.
Suppose $\mathit{u}(\mathit{x},t)={u}_{\mathrm{1}}(\mathit{x},t){\mathit{e}}_{\mathrm{1}}+{u}_{\mathrm{2}}(\mathit{x},t){\mathit{e}}_{\mathrm{2}}+{u}_{\mathrm{3}}(\mathit{x},t){\mathit{e}}_{\mathrm{3}}$ denotes the threedimensional velocity vector of the turbulent flow, where $\mathit{x}={x}_{\mathrm{1}}{\mathit{e}}_{\mathrm{1}}+{x}_{\mathrm{2}}{\mathit{e}}_{\mathrm{2}}+{x}_{\mathrm{3}}{\mathit{e}}_{\mathrm{3}}$ is the position and t is the time. We assume that the streamwise direction of the global mean flow U is in the direction of e_{1} such that U=Ue_{1} is (by definition) constant in space and time. We refer to e_{1} as the longitudinal direction and the components normal to that, i.e., e_{2} and e_{3}, as the transverse directions of the flow. As mentioned earlier, many experimental setups record only a onedimensional flow velocity at one location and as a function of time. We consider this onedimensional velocity time record to be in the longitudinal flow direction unless otherwise stated, e.g., when the probe misalignment is investigated. In the following, we first introduce different averaging principles that can be used to analyze turbulence statistics and Taylor's frozen hypothesis, and then we present the commonly used methods for extracting the energy dissipation rate. An introduction of the basic statistical description of turbulent flows is provided in the Appendix (Appendix A) for the sake of completeness.
2.1 On averaging, Reynolds decomposition and Taylor's hypothesis
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 onedimensional velocity time record should first be mapped onto a spatially resolved velocity field. This is achieved by invoking Taylor's hypothesis (Taylor, 1938), which requires a Reynolds decomposition of the velocity time record by separating the velocity fluctuations from the mean velocity. To perform the Reynolds decomposition, we first have to clarify what is meant by the mean velocity.
Generally, we have to distinguish between the global mean velocity $\mathit{U}=\langle \mathit{u}(\mathit{x},t)\rangle =U{\mathit{e}}_{\mathrm{1}}$, the volumeaveraged velocity 〈u(x,t)〉_{R} over a sphere of radius R (for onedimensional data 〈⋅〉_{R} denotes spatial averages over a window of size R), the timeaveraged velocity 〈u(x,t)〉_{τ} over a time interval τ and the ensembleaveraged velocity 〈u(x,t)〉_{N} over N realizations (Wyngaard, 2010; Pope, 2000, among others). In this work, 〈⋅〉 denotes the global mean, i.e., for infinitely large averaging windows in time or space. Thus, U is by definition independent of time and space, which in reality is valid only when u(x,t) is statistically stationary and homogeneous. Implicitly, $\langle \mathit{u}(\mathit{x},t){\rangle}_{R}=\mathrm{3}/\left(\mathrm{4}\mathit{\pi}{R}^{\mathrm{3}}\right){\iiint}_{\mathrm{0}}^{R}\mathrm{d}\mathit{x}\phantom{\rule{0.125em}{0ex}}\mathit{u}(\mathit{x},t)$ and $\langle \mathit{u}(\mathit{x},t){\rangle}_{\mathit{\tau}}=\frac{\mathrm{1}}{\mathit{\tau}}{\int}_{\mathit{\tau}/\mathrm{2}}^{\mathit{\tau}/\mathrm{2}}\mathrm{d}{t}^{\prime}\mathit{u}(\mathit{x},{t}^{\prime})$ are, respectively, local volume and time averages, as both R and τ are typically finite. In the limit of R, τ→∞, 〈u(x,t)〉_{R} and 〈u(x,t)〉_{τ} tend toward U. For repeatable experiments where identical experimental conditions are guaranteed, 〈u(x,t)〉_{N} tends toward U when N→∞.
The mean of a onedimensional velocity time record in the longitudinal direction U_{τ} here is defined by
such that the global mean $U={lim}_{\mathit{\tau}\to \mathrm{\infty}}{U}_{\mathit{\tau}}$, where τ is the averaging window. It should be noted that the global mean of the transverse velocity will be equal to zero; i.e., $\langle {u}_{\mathrm{2},\mathrm{3}}\left(t\right){\rangle}_{\mathit{\tau}}=\mathrm{0}$ when τ→∞, since here it is assumed that they are orthogonal to the mean flow direction. According to the Reynolds decomposition, the longitudinal velocity time record is composed of the mean velocity U and the random velocity fluctuation component ${u}_{\mathrm{1}}^{\prime}\left(t\right)={u}_{\mathrm{1}}\left(t\right)U$ so that the mean of the longitudinal velocity fluctuations $\langle {u}_{\mathrm{1}}^{\prime}\left(t\right)\rangle =\mathrm{0}$.
In certain circumstances, it is possible to map ${u}_{\mathrm{1}}^{\prime}\left(t\right)$ from time to space coordinates by applying Taylor's (frozeneddy) hypothesis (Taylor, 1938; Wyngaard, 2010), which relates temporal and spatial velocity statistics. Taylor (1938) argues that eddies can be regarded as frozen in time if they are passing the probing volume much faster than they evolve in time. This is the case if the turbulence intensity $I={\mathit{\sigma}}_{{u}_{\mathrm{1}}^{\prime}}/U$ is much smaller than unity, i.e., I≪1, where ${\mathit{\sigma}}_{{u}_{\mathrm{1}}^{\prime}}=\langle {u}_{\mathrm{1}}^{\prime \mathrm{2}}{\rangle}^{\mathrm{1}/\mathrm{2}}$ is the root mean square (rms) velocity fluctuation. Then, the series of time lags $\mathrm{\Delta}t=t{t}_{\mathrm{0}}$ relative to the start time t_{0} is mapped onto a distance vector with $\mathit{x}={\mathit{x}}_{\mathrm{0}}+U\mathrm{\Delta}t\phantom{\rule{0.125em}{0ex}}{\mathit{e}}_{\mathrm{1}}$ (Taylor, 1938), where x_{0} is the initial position at time t_{0}. This approach is found to be reliable for I≲0.25 (Nobach and Tropea, 2012; Wilczek et al., 2014; Risius et al., 2015), while it has been shown to fail when I>0.5 (Willis and Deardorff, 1976). The application of Taylor's hypothesis is inaccurate in the case of largescale variations in the velocity fluctuation field comparable with the mean velocity, which are known as “randomsweeping velocity” (Kraichnan, 1964; Tennekes, 1975). Complicating the estimation of the mean velocity, randomsweeping causes the mean energy dissipation rate to be consistently overestimated (Lumley, 1965; Wyngaard and Clifford, 1977).
One way to cope with nonstationary velocity time records is to evaluate the mean velocity for a subset of this signal. If the averaging time τ is finite, the time average U_{τ} may differ from the mean velocity U, causing a systematic bias in the subsequent data analysis. The estimation variance of the time average U_{τ} can be analytically expressed as (Wyngaard, 2010; Pope, 2000, among others)
where T is the integral timescale and $\langle {u}_{\mathrm{1}}^{\prime \mathrm{2}}\rangle $ the variance of the velocity time series. Notably, the size of the averaging window has to be large enough such that it fulfills $\langle {u}_{\mathrm{1}}^{\prime}\left(t\right){\rangle}_{\mathit{\tau}}\approx \mathrm{0}$ to apply the Reynolds decomposition.
2.2 Estimating the energy dissipation rate
The energy dissipation rate can be derived from various statistical quantities. A nonexhaustive list of the most common methods applicable to singlepoint measurements is shown in Table 1. Details of selected methods considered in this study are presented in the following subsections. If not explicitly mentioned, the averages denoted with 〈⋅〉 are defined globally.
2.2.1 Dissipative subrange
Proceeding from the Navier–Stokes equations for an incompressible Newtonian fluid, the instantaneous energy dissipation rate is given by 2ν(S_{ij}S_{ij}) (e.g., Pope, 2000; Davidson, 2015). As the velocity gradients are dominated by smallscale fluctuations, turbulent kinetic energy is dissipated into heat at small scales. Therefore, the contribution of largescale variations in the velocity is small compared with the contribution of the small scales (Pope, 2000; Elsner and Elsner, 1996). Hence, the instantaneous energy dissipation rate can be defined in terms of the velocity fluctuations only, i.e., replacing S_{ij} with the fluctuation strainrate tensor ${s}_{ij}=(\partial {u}_{i}^{\prime}(\mathit{x},t)/\partial {x}_{j}+\partial {u}_{j}^{\prime}(\mathit{x},t)/\partial {x}_{i})/\mathrm{2}$ (Pope, 2000):
Averaged over a sphere with radius R and volume 𝒱(R), the (local) volume average of the instantaneous energy dissipation rate is (Pope, 2000)
The local volume average ϵ_{R}(x,t) converges to the global mean energy dissipation rate if R tends toward infinity (Pope, 2000):
In experiments, it is often not possible to measure ϵ_{0}(x,t). Under the assumption of statistically homogeneous and isotropic turbulence, the volume and timeaveraged energy dissipation rate are typically inferred from onedimensional surrogates (Taylor, 1935; Elsner and Elsner, 1996; Siebert et al., 2006; Almalkie and de Bruyn Kops, 2012; Champagne, 1978; Donzis et al., 2008, among others), such as from the longitudinal velocity gradient (hence, the subscript G):
where the mapping between space and time domains is possible by applying Taylor's hypothesis if ${\mathit{\sigma}}_{{u}_{\mathrm{1}}^{\prime}}/U\ll \mathrm{1}$ and R_{11} is the longitudinal component of the velocity covariance tensor defined in Eq. (A1) (Siebert et al., 2006; Muschinski et al., 2004). The relationship shown in Eq. (6) is often called the “direct” method in the literature (e.g., Muschinski et al., 2004; Siebert et al., 2006) and requires a spatial resolution higher than the Kolmogorov length scale η_{K} to be accurate within ∼10 % (see Fig. A8).
2.2.2 Inertial subrange: structure functions
Kolmogorov's second similarity hypothesis from 1941 (Kolmogorov, 1941) provides another method for estimating the energy dissipation rate in the inertial range. Based on the inertialrange scaling of the nthorder longitudinal structure function, the mean energy dissipation rate can be calculated by (Pope, 2000)
where C_{n} is a constant, e.g., C_{2}≈2 (Pope, 2000), and ${\mathit{\zeta}}_{n}=n/\mathrm{3}$ according to K41 by dimensional analysis. In practice, ϵ_{I2} (Table 1) is retrieved by fitting either a constant to the compensated longitudinal secondorder structure function D_{LL}(r), n=2 in Eq. (7), or a power law ($\propto {r}^{\mathrm{2}/\mathrm{3}}$) to the inertial range of D_{LL}, defined in Eq. (A3), if the inertial range is pronounced over at least a decade. Accounting for intermittency, the scaling exponent of the nthorder structure function is modified to ${\mathit{\zeta}}_{n}=\frac{n}{\mathrm{3}}\left[\mathrm{1}\frac{\mathrm{1}}{\mathrm{6}}\mathit{\mu}(n\mathrm{3})\right]$, where μ is the internal intermittency exponent (Kolmogorov, 1962; Obukhov, 1962; Pope, 2000). 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 ${\mathit{\eta}}_{K}\ll r\ll L$. If the inertial range is not sufficiently pronounced, the extended selfsimilarity may be used to extend the inertial range (Benzi et al., 1993b, a). Otherwise, ϵ_{I2} can also be approximated by the maximum of Eq. (7) (for n=2) within the same range as before. This is possible because the maximum lies on the plateau in the case of a perfect K41 inertialrange scaling.
In the inertial range, the transverse secondorder structure function D_{NN}(r) is equal to $\mathrm{4}{D}_{\mathrm{LL}}\left(r\right)/\mathrm{3}$ in a coordinate system where r=re_{1} is parallel to the longitudinal flow direction (Pope, 2000), highlighting the importance of the measurement direction.
2.2.3 Inertial subrange: spectral method
According to K41 (Kolmogorov, 1941), the inertial subrange of the energy spectrum function scales as $E\left(\mathit{\kappa}\right)\propto \langle \mathit{\u03f5}{\rangle}^{\mathrm{2}/\mathrm{3}}{\mathit{\kappa}}^{\mathrm{5}/\mathrm{3}}$ with the wavenumber κ by dimensional analysis. In isotropic turbulence, the energy spectrum function can be converted into a onedimensional energy spectrum E_{11}(κ_{1}); see Eq. (A7). The wavenumber space is not directly accessible from onedimensional velocity time records. Relying on Taylor's hypothesis, the onedimensional energy spectrum E_{11}(κ_{1}) transforms into the frequency domain with ${F}_{\mathrm{11}}\left(f\right)=\mathrm{2}\mathit{\pi}{E}_{\mathrm{11}}\left({\mathit{\kappa}}_{\mathrm{1}}\right)/U$, where ${\mathit{\kappa}}_{\mathrm{1}}=\mathrm{2}\mathit{\pi}f/U$ (e.g., Wyngaard and Clifford, 1977; Oncley et al., 1996), yielding
which yields
with the Kolmogorov constant C_{K}=1.5 (Sreenivasan, 1995; Pope, 2000). 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 powerlaw spectra remains unaffected (Wyngaard and Clifford, 1977; Wilczek and Narita, 2012; Wilczek et al., 2014). Hence, Eq. (9) 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 of square meters per second (m^{2} s^{−1}), and $\langle {u}_{\mathrm{1}}^{\prime \mathrm{2}}\rangle ={\int}_{\mathrm{0}}^{\mathrm{\infty}}{F}_{\mathrm{11}}\left(f\right)\mathrm{d}f$. Under the assumption of Kolmogorov scaling in the inertial subrange, this identity can be adopted to estimate the mean energy dissipation rate from low and moderateresolution velocity measurements of a finite averaging window (Fairall et al., 1980; Siebert et al., 2006; O'Connor et al., 2010; Wacławczyk et al., 2017).
2.2.4 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., Lumley, 1992). In a dimensional argument, this rate is proportional to ${u}^{\mathrm{3}}\left(l\right)/l$, 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 ${\mathit{\sigma}}_{{u}_{\mathrm{1}}^{\prime}}$, the mean energy dissipation rate can be calculated by (Taylor, 1935)
where C_{ϵ} is the dissipation constant, and for time and spacevarying turbulence, it depends on both initial and boundary conditions as well as the largescale structure of the flow (Sreenivasan, 1998; Sreenivasan et al., 1995; Burattini et al., 2005; Vassilicos, 2015). C_{ϵ} is found to be about 0.5 for shear turbulence (Sreenivasan, 1998; Pearson et al., 2002) and 1.0 (Sreenivasan, 1984; Sreenivasan et al., 1995) or 0.73 (Sreenivasan, 1998) for grid turbulence. In this work, C_{ϵ} is assumed to be 0.5, which holds approximately in a variety of flows (Risius et al., 2015; Sreenivasan, 1995, and references therein).
Usually, the longitudinal integral length scale L_{11} is defined as (Pope, 2000)
where $f\left(r\right)={R}_{\mathrm{11}}\left(r{\mathit{e}}_{\mathrm{1}}\right)/{R}_{\mathrm{11}}\left(\mathbf{0}\right)$ is the longitudinal autocorrelation function (see also Eq. A1 and Table A1). However, due to experimental limitations, r_{0} is often given by the first zero crossing of f(r) in both laboratory and in situ measurements (e.g., Risius et al., 2015) or, alternatively, by the position where $f\left(r\right)=\mathrm{1}/$e (Tritton, 1977; Bewley et al., 2012). Griffin et al. (2019) carried out an integration for r→∞ by performing an exponential fit in the vicinity of $f\left(r\right)=\mathrm{1}/$e. Notably, ${E}_{\mathrm{11}}\left(\mathrm{0}\right)={\int}_{\mathrm{0}}^{\mathrm{\infty}}\mathrm{d}\mathit{\kappa}E\left(\mathit{\kappa}\right)/\mathit{\kappa}$ so that the estimation of L_{11} from the power spectrum is only recommended if $E\left(\mathit{\kappa}\right)=\frac{\mathrm{1}}{\mathrm{2}}{\mathit{\kappa}}^{\mathrm{3}}\frac{\mathrm{d}}{\mathrm{d}\mathit{\kappa}}\left(\frac{\mathrm{1}}{\mathit{\kappa}}\frac{\mathrm{d}{E}_{\mathrm{11}}\left(\mathit{\kappa}\right)}{\mathrm{d}\mathit{\kappa}}\right)$ (Pope, 2000) is accurately determined like in DNS. This approach requires not only a fully resolved velocity measurement but also a wellconverged 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.
2.3 Simulations of homogeneous isotropic turbulence
In this study, the direct numerical simulations of statistically homogeneous isotropic turbulent flow with $\mathrm{74}\le {R}_{\mathit{\lambda}}\le \mathrm{321}$ are used as the basis for evaluating the different methods for determining the dissipation rate (see Table 2). Thereby, the performance of the different methods in estimating the energy dissipation rate is not affected by violating fundamental assumptions, e.g., isotropy or homogeneity. The simulations are carried out with the parallel solver TurTLE (Lalescu et al., 2022), which solves the Navier–Stokes equations on a periodic domain using a pseudospectral method with a thirdorder Runge–Kutta time stepping. Here, we use a forcing scheme with a fixed energy injection rate on large scales.
To mimic an ensemble of singlepoint measurements, we introduced 1000 virtual probes into the flow (oneway 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 (e.g., Horst and Oncley, 2006; Freire et al., 2019) or noise (Lenschow and Kristensen, 1985; Antonia, 2003; Lewis et al., 2021). While the root mean square velocity fluctuation is determined by the Navier–Stokes simulation, we can control the mean flow speed through the speed of the virtual probe. The range of constant speeds used corresponds to turbulence intensities of 1 %–50 %. Along the trajectories, we then sample the local threedimensional velocity field (see Fig. 1) as well as the velocity gradient field, where we use spline interpolation to determine values in between grid points (Lalescu et al., 2010, 2022). By projecting the velocity vector onto the direction of the trajectory, e_{1}, and the orthogonal directions, e_{2} and e_{3}, we split the velocity field into longitudinal and transverse components, respectively. From the sampled velocity gradient tensor, we compute the local instantaneous dissipation ϵ_{0}. The time step is limited either by the stability requirements of the flow solver or, for smaller turbulence intensities, by the sampling frequency required to capture the underlying flow. Here, we choose the time step such that the distance traveled by the probe within one step is around 1$/$10 of the grid spacing, UΔt≈0.1Δx. The grid spacing Δx is chosen such that the highest resolved wavenumber k_{max} satisfies ${k}_{max}{\mathit{\eta}}_{K}\approx \mathrm{3}$.
Using Taylor's hypothesis, the longitudinal velocity time series correspond to at least 13L_{11} (for more details, see Table 2) so that second and thirdorder moments of both longitudinal velocity fluctuations and increments are reasonably converged (see Fig. A3). To estimate ϵ_{I3}, ϵ_{I2} and ϵ_{S}, the longitudinal structure functions are evaluated for scales $\mathrm{20}{\mathit{\eta}}_{K}\le r\le \mathrm{500}{\mathit{\eta}}_{K}$ or in the frequency domain for $\frac{U}{\mathrm{500}{\mathit{\eta}}_{K}}\le f\le \frac{U}{\mathrm{20}{\mathit{\eta}}_{K}}$. The groundtruth reference for the mean energy dissipation rate per virtual probe is given by 〈ϵ_{0}(x,t)〉_{VP}, i.e., the average of the dissipation field along the trajectory of each virtual probe. The global mean energy dissipation rate can be approximated by the ensemble average of all 〈ϵ_{0}(x,t)〉_{VP} from all virtual probes, i.e., 〈〈ϵ_{0}(x,t)〉_{VP}〉_{N}.
2.4 Variable Density Turbulence Tunnel (VDTT)
To evaluate the performance of different methods at Reynolds numbers applicable to atmospheric flows, we use the highresolution hotwire measurements of the longitudinal velocity components in the MPIDS VDTT (Bodenschatz et al., 2014). The VDTT datasets used here are associated with $\mathrm{147}\le {R}_{\mathit{\lambda}}\le \mathrm{5864}$, which enables us to bridge the gap between DNS ($\mathrm{74}\le {R}_{\mathit{\lambda}}\le \mathrm{321}$) and atmospheric R_{λ}∼𝒪(10^{3}) (Risius et al., 2015).
The VDTT is a recirculating wind tunnel where the working gas SF_{6} is pressurized up to 15 bar to increase its density, thereby enhancing the Taylorscale Reynolds number. To achieve the same density in air, the pressure would have had to be about 100 bar, which would have required a much thicker tunnel wall and more expensive construction. The VDTT has a horizontal length of 11.68 m and an inner diameter of 1.52 m where the rotation frequency of the fan sets the mean flow velocity ranging from 0.5 to 5.5 m s^{−1} (Bodenschatz et al., 2014). Longrange correlations of the turbulent flow determine its anisotropy. These longrange correlations are shaped with the help of an active grid consisting of 111 independently rotating winglets (Küchler et al., 2019; Küchler, 2021).
Longitudinal velocity fluctuations are temporally recorded with 30 to 60 µm long nanoscale thermal anemometry probes (NSTAPs; Bailey et al., 2010; Vallikivi et al., 2011, among others) or a 450 µm long conventional hot wire from Dantec (Jørgensen, 2001) corresponding to a resolution of <3η_{K} and <5η_{K}, respectively (Küchler et al., 2019), at variable distances from the active grid ranging from ≈ 6–9 m. The velocity measurements have been extensively characterized in terms of the mean flow profiles (Küchler, 2021) as well as the decay of turbulent kinetic energy (Sinhuber et al., 2015; Sinhuber, 2015), exposing velocity probability distribution functions (PDFs) as being flatter than Gaussian (Küchler, 2021). The inertialrange scaling exponent ζ_{2} of the longitudinal secondorder structure function is in agreement with Kolmogorov's revised phenomenology from 1962 (${\mathit{\zeta}}_{\mathrm{2}}=\mathrm{0.693}\pm \mathrm{0.003}$ for R_{λ}>2000) for a large variety of wake generation schemes (Küchler et al., 2020). In the case of hotwire measurements in the VDTT, the groundtruth energy dissipation rate for a given averaging window R is given by the gradient method 〈ϵ_{G}〉_{R}, which converges the fastest, as shown in Sect. 3.5.
2.5 Strategies for the evaluation of systematic and random errors
The virtual probes record onedimensional 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 〈ϵ〉_{R} estimates around the ground truth of the local mean energy dissipation rate defined in Eq. (4). The systematic error in the mean energy dissipation rate estimates expresses itself in a nonvanishing ensemble average of the deviations from the ground truth, i.e., the global volume average defined in Eq. (5).
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 R and I, respectively. One way to estimate these errors is to compare the estimated mean energy dissipation rate for a given averaging window R with the ground truth of the DNS defined by the mean energy dissipation rate per virtual probe track, i.e., $\langle {\mathit{\u03f5}}_{\mathrm{0}}(\mathit{x},t){\rangle}_{\mathrm{VP},\phantom{\rule{0.125em}{0ex}}R}$. Another possibility is to compare the estimates with the ensemble average of the mean energy dissipation rate from all virtual probes, i.e., $\langle \langle {\mathit{\u03f5}}_{\mathrm{0}}(\mathit{x},t){\rangle}_{\mathrm{VP},\phantom{\rule{0.125em}{0ex}}R}{\rangle}_{N}$, where N=1000 is the total number of virtual probes. Either of these possibilities is valid and would be interesting to understand. However, our analysis (not presented here) shows that the second approach is associated with a slightly higher value for the systematic error and a slightly higher standard deviation. For that reason, we have chosen the second approach to make a conservative assessment of the systematic errors; i.e., we compare the estimates of each method against $\langle \langle {\mathit{\u03f5}}_{\mathrm{0}}(\mathit{x},t){\rangle}_{\mathrm{VP},\phantom{\rule{0.125em}{0ex}}R}{\rangle}_{N}$ by
where $i\in \mathit{\{}G,I\mathrm{3},I\mathrm{2},S,L\mathit{\}}$ and 〈ϵ_{i}〉_{R} are the estimates of the energy dissipation rate via method i under the experimental limitation and imperfection such as the size of the averaging window or finite turbulence intensity. To distinguish between the different error terms in this paper, we refer to β as a “referencecompared” systematic error.
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 δ and refer to them as “selfcompared” errors. An experimental imperfection we considered here is the sensor misalignment: this is a nonzero angle of incidence θ between the true longitudinal flow direction along U and the one expected based on the sensor orientation. To investigate the isolated effect of sensor misalignment, we consider a specific set of DNSs with constant turbulence intensity (I=1 %) and the entire track length for each virtual probe. The selfcompared systematic error in each method due to misalignment is defined as
where ϵ_{i}(θ) is the estimate of the energy dissipation rate via method $i\in \mathit{\{}G,I\mathrm{3},I\mathrm{2},S,L\mathit{\}}$ from data with misalignment θ and ϵ_{i}(0) is the estimated dissipation rate from the same method and flow conditions but with an aligned sensor; i.e., θ=0.
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. (12) would be the best indicator of systematic errors. However, random errors due to the size of the averaging window can also be significant. When the spatial averaging window R (or temporal averaging window τ) is finite, we capture the selfcompared random error for each individual method by
where 〈ϵ_{i}〉_{R} is the local mean energy dissipation rate based on the averaging window R normalized by its ensemble average, i.e., 〈〈ϵ_{i}〉_{R}〉_{N}. Equation (14) indeed calculates the standard deviation of the normalized 〈ϵ_{i}〉_{R}, which is used here as a proxy for the random error. Table 3 provides an overview of the different error types and terminologies used here.
In the following, we first focus on the DNS data to calculate ϵ_{G}, ϵ_{I3}, ϵ_{I2}, ϵ_{S} and ϵ_{L} from the entire longitudinal velocity time records of all virtual probes and compare these estimates against the groundtruth reference. Then, we systematically investigate the impact of the turbulence intensity, (virtual) probe orientation and averaging window size for all methods of interest. The influence of the flow Reynolds number on the presented results is then discussed by taking into account the VDTT data together with the DNS data. Finally, we provide a proof of concept for a timedependent dissipation rate calculation by comparing the dissipation time series measured by ϵ_{G}, ϵ_{I2} and ϵ_{L} and its coarsegrained surrogate. In the following, we use the definitions of systematic and random errors as mentioned in Sect. 2.5 and Table 3.
3.1 Verification of the analytical methods and a first insight into their performance under ideal conditions
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_{I2} and ϵ_{I3} are obtained by a fit according to Eq. (7) with n=2 and n=3, respectively, in the inertial range with $r\in [\mathrm{20}{\mathit{\eta}}_{K},\mathrm{500}{\mathit{\eta}}_{K}]$ for DNS 2.1 and 3.1. Analogously, ϵ_{S} is inferred from the inertialrange fit in Eq. (9) in the range $f\in [U/(\mathrm{500}{\mathit{\eta}}_{K}),U/(\mathrm{20}{\mathit{\eta}}_{K}\left)\right]$. For DNS 1.1 with R_{λ}=74, due to the absence of an inertial range for low Taylorscale Reynolds numbers (see Fig. A7), the maximum of Eq. (7) is used to infer ϵ_{I2} and ϵ_{I3} instead of fitting the inertial range.
The distribution of the mean energy dissipation rate estimated by ${\mathit{\u03f5}}_{G},{\mathit{\u03f5}}_{I\mathrm{2}},{\mathit{\u03f5}}_{I\mathrm{3}},{\mathit{\u03f5}}_{S}$ and ϵ_{L} for each probe at R_{λ}=302 is shown in Fig. 2 and Table 4. Estimations for other R_{λ} values are shown in Fig. A1. The bestperforming method is the gradient method ϵ_{G}, with the range also being very close to the range of β_{ref}, whereas ϵ_{I3} is associated with the mean highest deviation. The superior performance of ϵ_{G} compared with the others is mainly due to the fact that it relies on (dissipationrange) secondorder statistics that can be captured with fast statistical convergence within a short sampling interval. Hence, the distributions of ϵ_{G} and ϵ_{ref} are similar. ϵ_{I3}, on the other hand, relies on thirdorder moments of the velocity increments of inertial scales associated with slower statistical convergence compared to ϵ_{G}. Therefore, ϵ_{I3} requires longer velocity records than ϵ_{G} to converge under stationary conditions. For this reason, the thirdorder 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 timedependent energy dissipation rate.
Figure 2 and Table 4 also show 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 secondorder quantities (in real and Fourier space, respectively) connected by f(r). Unlike ϵ_{G}, both ϵ_{I2} and, to a lesser extent, ϵ_{S} tend to overestimate the energy dissipation rate. However, ϵ_{S} more strongly depends on properly setting the fit range than ϵ_{I2} (Fig. A2). The spectral method ϵ_{S} can differ by a factor of 2 from ϵ_{I2} depending on the highfrequency limit. This factor of 2 is in accordance with a comparison of ϵ_{I2} and ϵ_{S} by a linear fit, resulting in a slope close to 0.5 (Akinlabi et al., 2019). 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 highfrequency limit of the inertial range of the spectrum is hardly distinguishable from its dissipation range, the choice of the fit range for ϵ_{S} is related to the fit range of the longitudinal secondorder structure function by $f\in \left[U/\left(\mathrm{500}{\mathit{\eta}}_{K}\right),U/\left(\mathrm{20}{\mathit{\eta}}_{K}\right)\right]$ as mentioned above. Wacławczyk et al. (2020) found that the estimation of the energy dissipation rate from the power spectral density is generally robust at small wavenumbers (i.e., larger length scales), whereas the secondorder structure function performs better at small length scales (i.e., larger wavenumbers). With our choice of the fit range $r\in [\mathrm{20}{\mathit{\eta}}_{K},\mathrm{500}{\mathit{\eta}}_{K}]$ for the DNS 3.1 dataset shown in Fig. 2, we confirm that ϵ_{I2} is already reliable at the lower end of the inertial range where dissipative effects are negligible.
Lastly, ϵ_{L} overestimates by 40 % on average. 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 in 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 due to the scatter in both ${\mathit{\sigma}}_{{u}_{\mathrm{1}}^{\prime}}$ and L_{11}, as illustrated in Figs. A3 and A4. However, the accuracy of the dissipation constant C_{ϵ}, which is a function of largescale forcing and initial conditions (Vassilicos, 2015; Sreenivasan, 1998; Sreenivasan et al., 1995; Burattini et al., 2005), can potentially cause larger mean deviations. Advantageously, the largescale estimate ϵ_{L} is applicable to lowresolution measurement. Figure A1 and Table A3 give an overview of the systematic errors in the different methods at different Reynolds numbers, showing that the above conclusions are also valid for lower R_{λ}.
3.2 The validity of Taylor's hypothesis and the impact of randomsweeping effects
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 3 shows the systematic error β_{i} for ϵ_{G}, ϵ_{I2}, ϵ_{S} and ϵ_{L} at different turbulence intensities for DNS 3.1–3.5. For each virtual probe taken into account in Fig. 3, we used the entire time series so that the size of the averaging window is maximal. While each method has a different systematic error and scatter, Fig. 3 indicates that the mean relative deviation of each estimate from the global mean 〈〈ϵ_{0}(x,t)〉_{VP}〉_{N} increases with turbulence intensity. This is particularly strong for the gradient method. For I=1 % and I=10 %, the gradient method has the lowest scatter in terms of the standard deviation ${\mathit{\sigma}}_{{\mathit{\beta}}_{G}}$ (19.3 % and 27.3 %) and the lowest systematic error in terms of 〈β_{G}〉_{N} (−0.5 % and 6.1 %), respectively. At higher turbulence intensities, ϵ_{I2} is the least affected method, with 〈β_{I2}〉_{N}=6.5 % and ${\mathit{\sigma}}_{{\mathit{\beta}}_{I\mathrm{2}}}=\mathrm{37.2}$ % for I=25 % as well as 〈β_{I2}〉_{N}=24.5 % and ${\mathit{\sigma}}_{{\mathit{\beta}}_{I\mathrm{2}}}=\mathrm{56.9}$ % for I=50 %. At the highest turbulence intensities, both ϵ_{L} and ϵ_{S} are associated with lower mean β than that of ϵ_{G}.
The fraction of track samples that can lead to a deviation of larger than 100 % increases from 0 % to ∼60 % for ϵ_{G} as the turbulence intensity increases from 1 % to 50 %. We hypothesize that these deviations of the mean are the result of randomsweeping effects, which limit the applicability of Taylor's hypothesis. In frequency space, Taylor's hypothesis (Taylor, 1938) establishes a onetoone mapping between the frequency and the streamwise wavenumber; i.e., ω=κ_{1}U. As the turbulence intensity grows, a randomly sweeping mean velocity smears out this correspondence between frequencies and wavenumbers. For the spectrum, this smearing out effectively moves energy from larger scales to smaller and less energetic ones (Lumley, 1965; Tennekes, 1975). Therefore, it leads to an overestimation in the inertial and dissipation range of the spectrum, thus affecting the inertial range and gradient method. To visualize this overestimation, we evaluate the effect of random sweeping on the spectrum (Eq. B2) numerically for different turbulence intensities at the example of a model spectrum (Eq. B4). The result is shown in Fig. 4, where the spectrum is premultiplied by ${\mathit{\kappa}}_{\mathrm{1}}^{\mathrm{2}}$ to later highlight the effect on the gradient method. Here, the overestimation is most pronounced in the dissipative range.
To quantify the impact of random sweeping on estimates of 〈ϵ〉, we first consider the influence of random sweeping on the gradient method. For the gradient method, Lumley (1965) and Wyngaard and Clifford (1977) have shown that in isotropic turbulence, random sweeping leads to an relative deviation of the volumeaveraged mean energy dissipation rate by
We illustrate this result in Appendix B, where we consider a model wavenumber–frequency spectrum (Wilczek and Narita, 2012; Wilczek et al., 2014), which is based on the same modeling assumptions used in Wyngaard and Clifford (1977). Due to the ${\mathit{\kappa}}_{\mathrm{1}}^{\mathrm{2}}$ weighting of the gradient method, the mean dissipation rate estimate is highly sensitive to the viscous cutoff of the energy spectrum, which is overestimated by randomsweeping effects (see Fig. 4). As a consequence, deviations in the estimated dissipation rate grow rapidly with turbulence intensity. In the right panel of Fig. 4, we compare the effect of random sweeping on the gradient method obtained with a model spectrum, the one computed by Lumley (1965) and the observed deviations by measurements of the virtual probes in a DNS flow; shown here are the DNS 3.1–3.5 cases. In fact, the estimate from Lumley (1965) can explain the magnitude of deviations observed by the virtual probes in the case of ϵ_{G} up to I=25 %. The strong deviation in β_{G} at I=50 % is likely due to the sensitivity of the gradients to the spacetotime conversion via Taylor's hypothesis: at high turbulence intensities, the mean velocity becomes smaller compared with the fluctuations. Therefore, the error in estimating the mean velocity due to the finite averaging window (Eq. 2) increases relative to the mean velocity. Larger relative errors in the estimated mean velocity lead – applying Taylor's hypothesis – to both under and overestimated spatial gradients for the individual averaging windows, in addition to the effect of random sweeping. Similarly this results in an additional overestimation of the dissipation rate. These deviations do not appear in evaluating randomsweeping effects based on a model spectrum, as there the mean velocity is a parameter we choose.
Now let us consider the two inertialsubrange methods. Here, as one can see in Figs. 3 and 4, the increase in the mean relative deviation, β_{i}, is less pronounced. In the inertial subrange, random sweeping causes an overestimation of the spectrum of merely several percent, while the inertialrange scaling is preserved as shown in Wyngaard and Clifford (1977) and Wilczek et al. (2014). As both the secondorder structure function and the spectral method are based on the inertial subrange of the energy spectrum, the effect of a randomly sweeping mean velocity on ϵ_{I2} and ϵ_{S} is expected to be small. Here, the overestimation of the spectrum can be used to express the relative systematic deviation in both ϵ_{I2} and ϵ_{S} for different turbulence intensities analytically:
where C_{T}(I) quantifies the spectral overestimation as a function of mean wind and fluctuations defined as in Wilczek et al. (2014). In Fig. 4b, we compare the observed deviations from the DNS to Eq. (16). This shows that Eq. (16) underestimates β_{I2} for $I\in \mathit{\{}\mathrm{0.01},\mathrm{0.05},\mathrm{0.1}\mathit{\}}$ (i.e., DNS 3.1, 3.2 and 3.3). The underestimation is most likely due to additional random errors associated with finite averaging window lengths. It is obvious from Table 2 that DNS 3.3 statistically has the shortest probe tracks of ∼3440η_{K} (DNS 3.1: ∼3550η_{K}, DNS 3.2: ∼3560η_{K}). Nonetheless, β_{I2} matches the prediction of Eq. (16) for $I\in \mathit{\{}\mathrm{0.25},\mathrm{0.5}\mathit{\}}$ where the corresponding probe tracks statistically amount to ∼5570η_{K} and ∼4260η_{K}, respectively. The effect of the averaging window size on ϵ_{I2} is explored in Sect. 3.4. We conclude that Eq. (16) can be used to estimate the error introduced by random sweeping of ϵ_{I2}.
For the spectral method, Eq. (16) underestimates the relative error β_{S} for all turbulence intensities. This may be due to the strong dependence of ϵ_{S} on the Ubased fitting range (see Fig. A2), i.e., $f\in \left[U/\left(\mathrm{500}{\mathit{\eta}}_{K}\right),U/\left(\mathrm{20}{\mathit{\eta}}_{K}\right)\right]$, which can differ significantly between virtual probes at high turbulence intensities. Further work is needed to assess the dependence of the spectral method on the choice of the fit range for finite turbulence intensities.
Overall, randomsweeping effects explain why the gradient method is more sensitive to turbulence intensity than inertialrange methods. Here, random sweeping accurately captures the deviations of the secondorder structure function method as a function of turbulence intensity, whereas it can only partially account for the observed deviations for the spectral method.
3.3 Probe misalignment
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 θ between the (virtual) anemometer and the global mean wind direction $\frac{\mathit{U}}{\left\mathit{U}\right}$ to be constant throughout the sampling trajectory. As can be seen from Eq. (C6), ϵ_{L} depends on θ. Then, the analytically derived error for ϵ_{L} due to misalignment of the sensor and the longitudinal wind direction is given by
where ϵ_{L}(θ) represents the energy dissipation that is derived given an angle of incidence θ, and ϵ_{L}(0) is the reference value for perfect alignment of the mean flow direction and the probe, i.e., when θ=0.
Analogously, the secondorder structure function tensor is affected by misalignment (cf. Appendix C). Thus, it can be shown that the analytically derived error δ_{I2}(θ) as a function of θ reads
where ϵ_{I2}(θ) represents the energy dissipation that is derived given an angle of incidence θ, and ϵ_{I2}(0) is the reference value for perfect alignment of the mean flow direction and the probe. As outlined in Appendix C and with Eq. (5), the analytically derived error in ϵ_{G} as a function of θ can be calculated to
where ϵ_{G}(θ) represents the energy dissipation that is derived given an angle of incidence θ, and ϵ_{G}(0) is the reference value for perfect alignment of the mean flow direction and the probe.
To compare the analytical expressions with DNS results, the sensing orientation of the virtual probes is rotated around the e_{3} axis in the coordinate system of each virtual probe by an angle θ relative to their direction of motion, i.e., the e_{1} axis. Then, ϵ_{L}(θ), ϵ_{I2}(θ) and ϵ_{G}(θ) are inferred from the new longitudinal velocity component. The ensembleaveraged relative errors in the estimated energy dissipation rates δ(θ) due to misalignment are shown as a function of θ in Fig. 5 in the range of ±50^{∘} both for DNS and for the analytically derived Eqs. (17)–(19). In general, the ensembleaveraged systematic errors follow the analytically derived errors reliably in terms of the limits of accuracy for all R_{λ} values at turbulence intensity I=1 %. The longitudinal secondorder structure function is the bestperforming method, with a systematic error 〈δ_{I2}〉_{N} of lower than 20 % for $\mathit{\theta}\in [\mathrm{25}{}^{\circ},\mathrm{25}{}^{\circ}]$, which increases to 100 % at $\mathit{\theta}=\pm \mathrm{50}$^{∘}. 〈δ_{L}〉_{N} is similarly affected by misalignment but is slightly larger than 〈δ_{I2}〉_{N}. Despite its rapid statistical convergence, ϵ_{G} is the method most vulnerable to misalignment compared with the other two methods.
In experiments where the sensor can be aligned to the mean wind direction within $\mathit{\theta}\in [\mathrm{10}{}^{\circ},\mathrm{10}{}^{\circ}]$ over the entire record time, δ_{i}(θ) is expected to be small. Further work is needed to evaluate the impact of a timedependent misalignment angle θ(t). We suppose that keeping the angle of attack θ fixed over the entire averaging window, here the entire time record of each probe, potentially leads to overestimation of δ_{i}(θ), with θ being a function of time in practice.
3.4 Systematic errors due to the finite averaging window of size R
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 R from the beginning of each track of virtual probes for the DNS 3.1 case. In this way, we obtain one subrecord for each virtual probe, which amounts to a total of 1000 subrecords for each averaging window R. From each of these subrecords, mean values of ϵ_{0} (i.e., $\langle {\mathit{\u03f5}}_{\mathrm{0}}(\mathit{x},t){\rangle}_{\mathrm{VP},\phantom{\rule{0.125em}{0ex}}R}$), 〈ϵ_{G}〉_{R}, 〈ϵ_{L}〉_{R} and 〈ϵ_{I2}〉_{R} are then evaluated. The smallest R considered for these analyses is 501η_{K}, which is limited by the upper bound of the fitting range $r\in [\mathrm{20}{\mathit{\eta}}_{K},\mathrm{500}{\mathit{\eta}}_{K}]$ for estimating ϵ_{I2}. The largest window size considered in this section is 3000η_{K}, which is limited by the total length of the virtual probe track (Table 2).
Before comparing estimates of the energy dissipation rate using different methods, let us first compare the locally averaged energy dissipation rate $\langle {\mathit{\u03f5}}_{\mathrm{0}}(\mathit{x},t){\rangle}_{\mathrm{VP},\phantom{\rule{0.125em}{0ex}}R}$ with the instantaneous energy dissipation rate, which is shown in Fig. 6a. All averaging window sizes create PDFs with similar shapes but significantly different from the shape of the instantaneous field. The larger the volume over which the dissipation field is averaged, the more the PDF$(\langle {\mathit{\u03f5}}_{\mathrm{0}}(\mathit{x},t\left){\rangle}_{\mathrm{VP},\phantom{\rule{0.125em}{0ex}}R}\right)$ converges to a peak at the global mean energy dissipation rate normalized by ${\dot{E}}_{\mathrm{in}}$, i.e., $\langle {\mathit{\u03f5}}_{\mathrm{0}}(\mathit{x},t)\rangle /{\dot{E}}_{\mathrm{in}}\approx \mathrm{1.0}$.
We can further explore the influence of the averaging window R for each method by examining the distribution of systematic errors, i.e., β_{i}, as shown in Fig. 6b–d. The first main point to note is the fact that at small R all methods tend to peak at a dissipation rate lower than the global. Hence, the mean energy dissipation rate is most likely underestimated. All PDF(β_{i}(R)) values become narrower, and the mean relative error β_{i}(R) converges to zero as R increases. The second main point to consider is the statistical uncertainty, causing a random error in estimating the local mean energy dissipation rate $\langle {\mathit{\u03f5}}_{\mathrm{0}}(\mathit{x},t){\rangle}_{\mathrm{VP},\phantom{\rule{0.125em}{0ex}}R}$. As can be seen in Fig. 6b–d, the width of the distribution is wide with asymmetric long tails, especially for β_{I2} and β_{L}. This is an indication that high random errors are to be expected in the estimation of the mean energy distribution rate.
3.5 Random errors due to the finite averaging window of size R
We now analytically focus on random errors associated with ϵ_{G}, ϵ_{L} and ϵ_{I2}. We denote 〈ϵ_{G}〉_{R}, 〈ϵ_{L}〉_{R} and 〈ϵ_{I2}〉_{R} as the energy dissipation rates that are estimated for a longitudinal velocity time record for a window of size R. For the calculation of random errors caused by the choice of the size of the averaging window, we consider DNS 1.3, 2.3 and 3.3, as well as wind tunnel experiments that all have a comparable turbulence intensity of I≈10 %.
Both the secondorder structure function in Eq. (A3) and the scaling argument in Eq. (10) depend on the variance $\langle {u}_{\mathrm{1}}^{\prime \mathrm{2}}\rangle $ of the longitudinal velocity time record. ϵ_{G} is also related to $\langle {u}_{\mathrm{1}}^{\prime \mathrm{2}}\rangle $ through Eqs. (6) and (A1). The variance $\langle {u}_{\mathrm{1}}^{\prime \mathrm{2}}\rangle $ itself is subject to both systematic and random errors in the case of a finite averaging window R<∞. Assuming an ergodic and hence stationary velocity fluctuation time record with a vanishing mean, the systematic error in estimating the variance over an averaging window of size R is given by (following Lenschow et al., 1994, while applying Taylor's hypothesis)
where $\langle {u}_{\mathrm{1}}^{\prime \mathrm{2}}{\rangle}_{R}$ is the estimated variance based on the (finite) averaging window R, $\langle {u}_{\mathrm{1}}^{\prime \mathrm{2}}\rangle $ is the true variance and it is assumed R≫L_{11}. The always negative error predicted by Eq. (20) indicates that, for finite averaging window sizes, the variance $\langle {u}_{\mathrm{1}}^{\prime \mathrm{2}}\rangle $ is always statistically underestimated, which agrees with Fig. A3a. Equation (20) furthermore indicates that the systematic error in the variance estimates can be neglected for sufficiently long averaging windows R≫L_{11}.
The variance estimates are also subject to statistical uncertainty, which is also known as the random error in variance estimation (Lenschow et al., 1994). Assuming that ${u}_{\mathrm{1}}^{\prime}\left(t\right)$, which has a zero mean, can be modeled by a stationary Gaussian process and that its autocorrelation function is sufficiently well represented by an exponential, the random error in estimating the variance can be expressed as (following Lenschow et al., 1994, while applying Taylor's hypothesis)
where it is assumed R≫L_{11} such that the systematic error can be neglected and, hence, ${\u2329\langle {u}_{\mathrm{1}}^{\prime \mathrm{2}}{\rangle}_{R}\u232a}_{N}\approx \langle {u}_{\mathrm{1}}^{\prime \mathrm{2}}\rangle $. Here, ${\u2329\langle {u}_{\mathrm{1}}^{\prime \mathrm{2}}{\rangle}_{R}\u232a}_{N}$ is the ensemble average of the variance estimates $\langle {u}_{\mathrm{1}}^{\prime \mathrm{2}}{\rangle}_{R}$ for an averaging window R. It can be seen that e_{rand} is larger than the systematic error in Eq. (20) when R>L_{11}.
Consequently, the estimation of the mean energy dissipation rate by the scaling argument in Eq. (10) is affected by the (absolute) random error in the variance estimation given by the product of e_{rand} and ${\u2329\langle {u}_{\mathrm{1}}^{\prime \mathrm{2}}{\rangle}_{R}\u232a}_{N}$. Invoking the Gaussian error propagation, the analytically estimated error reads
where e_{rand} is the relative random error in the variance estimate of the velocity fluctuations $\langle {u}_{\mathrm{1}}^{\prime \mathrm{2}}{\rangle}_{R}$ defined in Eq. (21), and $\langle {u}_{\mathrm{1}}^{\prime \mathrm{2}}{\rangle}_{R}$ is the variance estimate of ${u}_{\mathrm{1}}^{\prime}$ based on the averaging window R. Then, the absolute random error in the variance estimate of the velocity fluctuations $\langle {u}_{\mathrm{1}}^{\prime \mathrm{2}}{\rangle}_{R}$ is given by ${e}_{\mathrm{rand}}\langle {u}_{\mathrm{1}}^{\prime \mathrm{2}}{\rangle}_{R}$. δ_{L}(R) is a relative error itself, hence the prefactor $\mathrm{1}/\langle {\mathit{\u03f5}}_{L}{\rangle}_{R}$. Notably, δ_{L}(R) scales as ${R}^{\mathrm{1}/\mathrm{2}}$.
Similarly, the longitudinal secondorder structure function is also affected by the estimation variance $\langle {{u}_{\mathrm{1}}^{\prime}}^{\mathrm{2}}{\rangle}_{R}$,
where D_{LL}(r;R) is the longitudinal secondorder structure function evaluated over an averaging window of size R and under the assumption that the longitudinal autocorrelation function f(r) is sufficiently converged over the range of the averaging window. This is a simplistic assumption that may be questionable in some cases, but a more robust evaluation of the validity of the assumption is complex and beyond the scope of this study.
Thus, the uncertainty in estimating the variance propagates to 〈ϵ_{I2}〉_{R} relying on D_{LL}(r;R) (Eq. 7 for n=2). The random error δ_{I2}(R) can be analytically inferred from the random error in the secondorder structure function ${\mathit{\sigma}}_{{D}_{\mathrm{LL}}}$ by Gaussian error propagation, yielding
which shows that δ_{I2}(R) scales as ${R}^{\mathrm{1}/\mathrm{2}}$ similar to δ_{L}(R). Considering Eqs. (6) and (A1), the gradient method can also be expressed as a function of the variance $\langle {u}_{\mathrm{1}}^{\prime \mathrm{2}}\rangle $. Hence, Gaussian error propagation yields
assuming R≫L_{11} such that the systematic error is negligible so that ${\u2329\langle {u}_{\mathrm{1}}^{\prime \mathrm{2}}{\rangle}_{R}\u232a}_{N}\approx \langle {u}_{\mathrm{1}}^{\prime \mathrm{2}}\rangle $.
Equations (22), (24) and (25) are expressed as a function of R and L_{11}, which do not reveal the dependency of random errors on the Reynolds number. In addition, this expression relies on large scales that depend on the scale of the energy injection, which makes it difficult to fairly compare the errors between different flows as it is not a universal feature. Therefore, we want to link the averaging window to the Kolmogorov length scale η_{K}, which only depends on the viscosity and the mean energy dissipation rate. We can rewrite these equations in terms of η_{K}, R and R_{λ} as follows:
where we have invoked ${L}_{\mathrm{11}}/L\sim \mathrm{1}/\mathrm{2}$, which is valid at sufficiently high R_{λ}, and have used the relationship $L/{\mathit{\eta}}_{K}={\left(\frac{\mathrm{3}}{\mathrm{20}}{R}_{\mathit{\lambda}}^{\mathrm{2}}\right)}^{\mathrm{3}/\mathrm{4}}$ (Pope, 2000). Following the intuition, the longer the averaging window, the smaller the random error in each method.
Furthermore, Eqs. (26) and (27) provide means to choose a suitable averaging window size to achieve a given randomerror threshold a. Let R_{a} be the averaging window of size R such that δ_{i}(R)<a. Then, the required averaging window R_{a} for ϵ_{I2} and ϵ_{L} is
where the required averaging window size R_{a} scales with ${R}_{\mathit{\lambda}}^{\mathrm{3}/\mathrm{2}}$. Similarly, the required averaging window for ϵ_{G} is
For example, for the random errors of ϵ_{I2} and ϵ_{L} to be less than 10 % at R_{λ}=1000, the averaging window should be $R\sim \mathrm{2}\times {\mathrm{10}}^{\mathrm{6}}{\mathit{\eta}}_{K}\sim \mathrm{2}\times {\mathrm{10}}^{\mathrm{4}}{L}_{\mathrm{11}}$, while for ϵ_{G} the required averaging window is $R\sim \mathrm{8}\times {\mathrm{10}}^{\mathrm{5}}{\mathit{\eta}}_{K}\sim {\mathrm{10}}^{\mathrm{4}}{L}_{\mathrm{11}}$.
Figure 7 shows the empirical random errors δ_{G}(R) (Fig. 7a) and δ_{I2}(R) (Fig. 7b) as a function of the averaging window size for various R_{λ} values based on VDTT data (for ϵ_{L}, see Fig. A5). To do this, we select an averaging window of size R, where $\mathrm{1000}{\mathit{\eta}}_{K}<R<\mathcal{O}\left({\mathrm{10}}^{\mathrm{6}}{\mathit{\eta}}_{K}\right)$, from the beginning of each 30 s time segment of the VDTT longitudinal velocity records (a total of 47 to 597 time segments depending on R_{λ}).
The scaling of δ_{G}(R) and δ_{I2}(R) is predicted well for R≳10L_{11} as expected from Eqs. (25) and (24) and the assumptions we made to derive them. However, for smaller R a statistical convergence of ϵ_{G}, ϵ_{I2} or ϵ_{L} against the mean energy dissipation rate cannot be expected, in particular if $R/{L}_{\mathrm{11}}<\mathrm{1}$.
Furthermore, it is evident from Fig. 7 that the random errors do not fully collapse on each other for different Reynolds numbers and at a given $R/{L}_{\mathrm{11}}$. Moving horizontally on a line of constant random error, e.g., the dashed line of 50 % error, the required window size increases with R_{λ}, as shown in the insets of Fig. 7a and b. Predictions of Eqs. (28) and (29) are also shown in these plots via solid blue lines.
For both ϵ_{G} and ϵ_{I2}, the theoretical expectation for R_{a} tends to overestimate the actual averaging window size at which a random error of 50 % is achieved. This overestimation is expected as the theoretical expectation for R_{a} in Eqs. (28) and (29) is derived assuming that largescale quantities such as f(r) and L_{11} are fully converged. However, ϵ_{G} is technically relying on small scales. ϵ_{G} depends on velocity fluctuation gradients, which are numerically obtained by central differences. Hence, each increment in the velocity record contributes to the average in the gradient method (Eq. 6). In the case of ϵ_{I2}, the number of possible increments reduces for larger separations for a finite averaging window. By definition, the exact computation of L_{11} even requires a fully converged f(r) for all r values.
However, VDTT experiments with R_{λ}>3000 underestimate the prediction of Eq. (20) by about a factor of 2. This is particularly clear for ϵ_{L} shown in Fig. A5. This 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., Eqs. (24), (22) and (25). In particular, for experiments with high Re in VDTT, the assumption of Gaussian velocity fluctuations with zero skewness is questionable, as shown in Fig. A6. Lenschow et al. (1994) have already established that the size of the averaging window for a skewed Gaussian process (see Eq. 19 in Lenschow et al., 1994) 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.
3.6 Estimating the transient energy dissipation rate
As has been shown previously in Figs. 6 and 7, both systematic and random errors decrease with the size of the averaging window. For a correct estimate of the magnitude, it is therefore advantageous to choose the averaging window as large as possible, but the price of this is that the transient trend smaller than the selected window size cannot be reproduced. In addition, it is also important to know to what extent statistical uncertainties originating from the estimation methods themselves disguise the true trends of the underlying turbulent flow. Given a certain averaging window size R, here, we empirically evaluate if trends in the coarsegrained time series are physical or rather statistical. In other words, we ask the question whether local estimates of the mean energy dissipation rate follow the groundtruth reference $\langle {\mathit{\u03f5}}_{\mathrm{0}}(\mathit{x},t){\rangle}_{\mathrm{VP},\phantom{\rule{0.125em}{0ex}}R}$ or not. Respecting the intermittent nature of turbulence and energy dissipation, the standard deviation of $\langle {\mathit{\u03f5}}_{\mathrm{0}}(\mathit{x},t){\rangle}_{\mathrm{VP},\phantom{\rule{0.125em}{0ex}}R}$ is a first proxy for the variability of the trend in $\langle {\mathit{\u03f5}}_{\mathrm{0}}(\mathit{x},t){\rangle}_{\mathrm{VP},\phantom{\rule{0.125em}{0ex}}R}$. Hence, detecting the true trend requires that β_{i} and δ_{i}(R) are smaller than the standard deviation of $\langle {\mathit{\u03f5}}_{\mathrm{0}}(\mathit{x},t){\rangle}_{\mathrm{VP},\phantom{\rule{0.125em}{0ex}}R}$.
It can already be concluded from Figs. 2, 7, A1 and A5 that ϵ_{G} is the most promising candidate to capture the true trend. However, to fully answer the above questions, we need to conduct a more indepth analysis. The upper plot in Fig. 8 shows the rescaled and coarsegrained dissipation field $\langle {\mathit{\u03f5}}_{\mathrm{0}}(\mathit{x},t){\rangle}_{\mathrm{VP},\phantom{\rule{0.125em}{0ex}}R}$ for a sliding window of size R≈5500η_{K} and a turbulence intensity I=10 % obtained from the time series of one virtual probe for case DNS 2.0 (“probe 0”). Consistent with results shown earlier, 〈ϵ_{G}〉_{R} follows $\langle {\mathit{\u03f5}}_{\mathrm{0}}(\mathit{x},t){\rangle}_{\mathrm{VP},\phantom{\rule{0.125em}{0ex}}R}$ best in comparison with 〈ϵ_{I2}〉_{R} and 〈ϵ_{L}〉_{R}. Both 〈ϵ_{I2}〉_{R} and 〈ϵ_{L}〉_{R} are associated with substantial scatter, although 〈ϵ_{I2}〉_{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. 8 shows 〈ϵ_{I2}〉_{R} together with the random error of ϵ_{I2} as defined by Eq. (24). Despite the strong scatter, the groundtruth reference is nearly always within the error bar of ϵ_{I2}, with some exceptions, e.g., $r/{\mathit{\eta}}_{K}<\mathrm{5000}$ or $r/{\mathit{\eta}}_{K}\approx $ 60 000. It can also be seen that 〈ϵ_{I2}〉_{R} is, if at all, only weakly correlated with the groundtruth reference $\langle {\mathit{\u03f5}}_{\mathrm{0}}(\mathit{x},t){\rangle}_{\mathrm{VP},\phantom{\rule{0.125em}{0ex}}R}$ for a window size of $R/{\mathit{\eta}}_{K}\approx \mathrm{5500}$. This shows that it is extremely difficult, if at all possible, to track the true trend with lowresolution time records.
To assess this correlation more quantitatively, we evaluate the Pearson correlation coefficient between the groundtruth reference $\langle {\mathit{\u03f5}}_{\mathrm{0}}(\mathit{x},t){\rangle}_{\mathrm{VP},\phantom{\rule{0.125em}{0ex}}R}$ and ϵ_{G}, ϵ_{I2} as well as ϵ_{L}, respectively, as a function of the rescaled averaging window size $R/{\mathit{\eta}}_{K}$ for all virtual probes of case DNS 2.0. As an example, the Pearson correlation coefficient between ϵ_{0}(x,t)〉_{R} and ϵ_{I2} is 0.33 in Fig. 8 (upper plot). Figure 9a shows the ensemble averages of the Pearson correlation coefficient together with the standard error (shaded area). While 〈ϵ_{G}〉_{R} has a pronounced correlation with the groundtruth reference $\langle {\mathit{\u03f5}}_{\mathrm{0}}(\mathit{x},t){\rangle}_{\mathrm{VP},\phantom{\rule{0.125em}{0ex}}R}$, both 〈ϵ_{I2}〉_{R} and 〈ϵ_{L}〉_{R} are only very weakly correlated with 〈ϵ_{G}〉_{R}.
The effect of R_{λ} on the Pearson correlation coefficient is also shown in Fig. 9b for the VDTT experiments at various R_{λ} values. Here, we compare ϵ_{I2} and ϵ_{L} to ϵ_{G} in the absence of a groundtruth reference. To ensure a negligible systematic error, we chose a fixed averaging window of R=30L_{11} for each R_{λ}. Figure 9b shows that the correlation for ϵ_{I2} is always higher than that of ϵ_{L}, except for very low R_{λ}. There is a nonmonotonic behavior in the correlation coefficients in Fig. 9b that seems to be related to the skewness values shown in Fig. A6. Nonetheless, there is a clear increase in correlation coefficients with R_{λ}. Firstly, the random error in δ_{I2}(R) ranges from 20 % to 40 % at R=30L_{11}. Secondly, the kurtosis of the instantaneous energy dissipation field scales with ${R}_{\mathit{\lambda}}^{\mathrm{3}/\mathrm{2}}$ (Pope, 2000), which is why the variability in the instantaneous energy dissipation field increases with R_{λ}. Hence, at small ${R}_{\mathit{\lambda}}^{\mathrm{3}/\mathrm{2}}$ and R=30L_{11}, $\langle {\mathit{\u03f5}}_{I\mathrm{2}}{\rangle}_{\mathrm{30}{L}_{\mathrm{11}}}$ scatters only randomly around the global mean energy dissipation rate (with a 3 % standard deviation of $\langle {\mathit{\u03f5}}_{G}{\rangle}_{\mathrm{30}{L}_{\mathrm{11}}}$), which is why the correlation coefficient is low. In contrast, at large R_{λ} and R=30L_{11}, the locally averaged mean energy dissipation rate $\langle {\mathit{\u03f5}}_{G}{\rangle}_{\mathrm{30}{L}_{\mathrm{11}}}$ fluctuates more strongly (≈30 % standard deviation of $\langle {\mathit{\u03f5}}_{G}{\rangle}_{\mathrm{30}{L}_{\mathrm{11}}}$) where δ_{I2}(R) is already comparable.
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 onedimensional 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 inertialrange methods due to randomsweeping effects. Low values of turbulence intensity and ideal alignment of probes can be best controlled in groundbased measurements. Measurements aboard research aircraft traveling at a true speed of 100 m s^{−1} can also satisfy these conditions, but the spatial and temporal resolution required to measure the velocity gradients (see Fig. A8) is challenging to achieve; i.e., a robust probe with a wire length ideally smaller than 1 mm (or of the same order as η_{K}) and a true response frequency of 10^{5} Hz are needed. Other airborne platforms, such as helicopters, balloons and kites, would encounter higher turbulence intensities, making Taylor's frozen flow assumption more difficult to satisfy. Most importantly, however, they all suffer from probe alignment into the local wind at the scale of the measurement platform.
Other estimates based on inertialsubrange methods (cf. Table 1) are less sensitive to a high turbulence intensity and probe alignment. Regarding the impact of a high turbulence intensity, we find the most reliable method for most field applications to be the secondorder structure function. However the accuracy is at least 10 %, even in most ideal conditions considered here (Table 4). Considering experimental imperfections, the actual deviation is expected to be of the order of 100 % (Figs. 3 and 5).
Given atmospheric turbulence with η_{K}∼1 mm and L_{11}∼100 m, the inertial range of the secondorder structure function extends from about 60η_{K} (Pope, 2000) to at most the integral scale, if at all. However, for applying the secondorder SF method (inertial range), at least 1 to 2 decades of the inertial range are needed. Assuming that the fitting range is chosen from 0.1 to 10 m, a measurement platform with a true air speed of 10–100 m s^{−1} requires an anemometer with a sampling frequency of 100–1000 Hz in order to provide about 2 decades of data within the inertial range.
Our analysis further shows that estimating the transient energy dissipation in atmospheric clouds (L_{11}∼100 m, 𝒪(R_{λ})∼10^{3}–10^{4}) via the secondorder structure function (inertial range) with an averaging window of R=100 m is prone to random errors of the order of 100 % (Fig. 7b). Shorter averaging windows involve even higher random errors. It is therefore recommended to choose the averaging window as large as possible but still smaller than length and timescales on which the atmosphere remains homogeneous and stationary, respectively. This recommendation is not easy to achieve in an atmosphere. For example, measurements in the marine boundary layer with intermittent shallow cumulus clouds require a careful consideration in regard to stationary conditions. Then, two approaches can be recommended. On the one hand, one can choose a very long averaging window to average over many individual clouds associated with a small random error but potentially violating the stationary and homogeneity condition. On the other hand, one could investigate the transient mean energy dissipation rate (as a proxy for the local mean energy dissipation rate per cloud), while accepting a large random error.
The large random errors also preclude the possibility of interpreting trends in the measured energy dissipation rate via the secondorder structure function if we consider the results shown in Figs. 8 and 9a. Even if we consider the promising correlations shown in Fig. 9b at high Reynolds numbers, interpreting trends and patterns in atmospheric data from onedimensional time records remains a challenge, especially in an atmosphere with intermittent cloudiness. As an example, in a cloud with a horizontal extent of 1 km, one can obtain 10 nonoverlapping data blocks with a window length of 100 m, which corresponds to a random error of about 100 % to be combined with a systematic error of the same order of magnitude. With 10 data points combined with such large uncertainties, it would be extremely difficult to interpret the change in the dissipation rate in this cloud. Interpretations of trends in the energy dissipation rate would be particularly flawed if the size of the averaging window is <10L_{11}. At best, one average value can be determined for such small clouds, which would still deviate from the global mean by at least 50 %–100 % if (and this is a big if) the cloud is sufficiently stationary and homogeneous for averaging. However, the limitations imposed by the validity of the Taylor hypothesis can be mitigated by using an array of highresolution anemometers. The array should be oriented so that the anemometers point in the mean local wind direction. The calculation of transverse velocity gradients or secondorder structure functions would no longer depend on the applicability of the Taylor hypothesis, since the distances between the sensors are predefined by the design.
We have presented an extensive review on the analysis procedure for estimating the energy dissipation rate from singlepoint onedimensional velocity time records along with an overview of the advantages and disadvantages (see Table 1). To conclude, this paper provides means to estimate errors in the global and local mean energy dissipation rates due to experimental imperfections and limitations. These estimated errors can be used to assess the quality and accuracy of the measurement. Furthermore, error estimates of the global and local mean energy dissipation rates can be used to assess the errors in other turbulence and cloud droplet parameters, e.g., various turbulence length scales and the cloud droplet Stokes number, with the help of Gaussian error propagation. A set of practical guidelines for measurement and analysis strategies has been provided in the previous section, and the following presents a technical summary of the main results.
The main methods considered in this study are the gradient method ϵ_{G}, the secondorder SF (inertialrange) method ϵ_{I2}, the spectral method ϵ_{S} and the scaling argument ϵ_{L}. We have provided a systematic assessment of the accuracy of inferring the energy dissipation rate from such onedimensional velocity time series as a function of turbulence intensity, probe orientation with respect to the longitudinal direction and the effect of a finite averaging window size. We used DNS data with Reynolds numbers in the range $\mathrm{74}\le {R}_{\mathit{\lambda}}\le \mathrm{321}$ as well as experimental data from highresolution onedimensional wind tunnel measurements with Reynolds numbers in the range $\mathrm{147}\le {R}_{\mathit{\lambda}}\le \mathrm{5864}$ to evaluate the performance of different methods against robust benchmark values. The results presented in this study help to assess the accuracy of the energydissipationrate estimates as a function of several parameters, such as finite turbulence intensity, misalignment between sensor and longitudinal flow direction, and finite size of the averaging window. The main results are as follows:

Each method could reproduce the groundtruth reference 〈ϵ(x,t)〉 to within less than 10 % for wellconverged statistics and at low turbulence intensity. The most accurate method is the gradient method (ϵ_{G}), and the least accurate method is the one based on the $\mathrm{4}/\mathrm{5}$ law (ϵ_{I3}) (see Fig. 2). The referencecompared systematic error tends to be overestimated due to the global choice of the fit range; e.g., lower systematic errors for ϵ_{I2} can be obtained by choosing a fit range for each DNS dataset that is in a range where the scaling of the structure function is closest to the expected scaling.

In the case of finite turbulence intensities, ϵ_{G}, ϵ_{S} and ϵ_{I2} systematically overestimate the groundtruth energy dissipation rate. The gradient method (ϵ_{G}) is the most affected by a finite turbulence intensity I, whereas ϵ_{I2} is the least affected (see Figs. 3 and 4b). The overestimation can be captured by a randomadvection model (Fig. 4). Regarding the smallscale estimate ϵ_{G}, the error formula provided by Lumley (1965) (β_{G}∝5I^{2}) captures the effect of random advection.

Considering the probe orientation, the gradient method (ϵ_{G}) is the most affected by misalignment between the probe orientation and the longitudinal flow direction, whereas ϵ_{I2} is the least affected (Fig. 5) (compare Eqs. 19, 17 and 18).

We provide scaling arguments δ_{i}(R) to estimate the required averaging window size optimized for a desired randomerror threshold for ϵ_{G} in Eq. (29), ϵ_{I2} in Eq. (28) and ϵ_{L} in Eq. (28). With this, we can estimate a coarsegrained energy dissipation rate to within a predicted uncertainty as shown in Fig. 8. Systematic errors β_{i} are smaller than random errors δ_{i}(R) for R>2L_{11}.

The random error in the gradient method δ_{G}(R) converges at least 4–5 times faster than ϵ_{I2} (compare Eqs. 28 and 29).

Only ϵ_{G} estimates the transient energy dissipation rate 〈ϵ_{0}〉_{R} reliably, although it is the most vulnerable to experimental imperfections/limitations.
As discussed in detail below, the mean energy dissipation rate can be related to secondorder statistics of the velocity field, either in terms of velocity gradients or in terms of velocity increments. In any case, the twopoint velocity covariance tensor turns out to be the central quantity of interest from which the secondorder 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 twopoint quantities depend only on the separation vector r. All averages are invariant under rotations of the coordinate system, and the mean squared velocity fluctuation is identical for all velocity components; i.e., $\langle {u}^{\prime \mathrm{2}}\rangle =\langle {u}_{\mathrm{1}}^{\prime \mathrm{2}}\rangle =\langle {u}_{\mathrm{2}}^{\prime \mathrm{2}}\rangle =\langle {u}_{\mathrm{3}}^{\prime \mathrm{2}}\rangle $. We provide an overview of the most relevant definitions, their notations and their conventions. This section does not explicitly discuss the effect of the averaging window, but the definitions presented can be applied to windowed inputs with no or straightforward modifications.
Under the given assumptions, the twopoint velocity covariance tensor takes the form (e.g., Pope, 2000; Robertson, 1940; Batchelor, 1953)
where $f\left(r\right)={R}_{\mathrm{11}}\left(r\right)/{R}_{\mathrm{11}}\left(\mathrm{0}\right)$ and $g\left(r\right)=f\left(r\right)+r{\partial}_{r}f\left(r\right)/\mathrm{2}$ are the longitudinal and transverse autocorrelation functions, respectively, with $f\left(\mathrm{0}\right)=g\left(\mathrm{0}\right)=\mathrm{1}$. Notably, if one chooses r=re_{1}, then ${R}_{\mathrm{11}}\left(r\right)=\langle {u}^{\prime \mathrm{2}}\rangle f\left(r\right)$ and ${R}_{\mathrm{22}}\left(r\right)={R}_{\mathrm{33}}\left(r\right)=\langle {u}^{\prime \mathrm{2}}\rangle g\left(r\right)$ as well as all other components vanish (e.g., Pope, 2000). As a remarkable consequence, R_{ij}(r) is uniquely defined by f(r) in isotropic turbulence. As mentioned below, the integral length scale as well as the Taylor microscale are determined by f(r) (Pope, 2000).
Analogously, a covariance tensor can be defined for velocity increments, i.e., the secondorder velocity structure function tensor (Pope, 2000; Davidson, 2015)
The longitudinal secondorder structure function D_{11}(r) is related to f(r) by (e.g., Pope, 2000; Davidson, 2015)
As explained below, by measuring the longitudinal secondorder structure function D_{LL}(r), the mean energy dissipation rate can be inferred from the inertialrange scaling of the longitudinal structure function (cf. Eq. 7).
Furthermore, the velocity gradient covariance tensor can also be defined in terms of the velocity covariance tensor
Since the local and instantaneous energy dissipation rate (cf. Eq. 3) is defined in terms of the strainrate tensor ${S}_{ik}=\left(\partial {u}_{i}^{\prime}(\mathit{x},t)/\partial {x}_{k}+\partial {u}_{k}^{\prime}(\mathit{x},t)/\partial {x}_{i}\right)/\mathrm{2}$, the mean energy dissipation rate can be directly related to contractions of the velocity gradient covariance tensor evaluated at zero. Note that in a turbulent flow with a zero mean velocity, the strainrate tensor S_{ik} is equal to the fluctuation strainrate tensor s_{ik}.
The twopoint velocity covariance tensor can be expressed in Fourier space through the energy spectrum tensor (Pope, 2000)
where κ is the wave vector. For SHI turbulence, Φ_{ij}(κ) takes the form
where E(κ) is the energy spectrum function.
Since access to the full energy spectrum function is not always available, onedimensional spectra are of interest, too. The mean energy dissipation rate can be estimated from the inertialrange scaling of the longitudinal onedimensional spectrum (as shown in Eq. 9), which can be calculated by both the energy spectrum function and the velocity covariance tensor (Pope, 2000)
with the wavenumber κ_{1} corresponding to the e_{1} direction and ${R}_{\mathrm{11}}\left(\mathrm{0}\right)=\langle {u}^{\prime \mathrm{2}}\rangle ={\int}_{\mathrm{0}}^{\mathrm{\infty}}{E}_{\mathrm{11}}\left({\mathit{\kappa}}_{\mathrm{1}}\right)\mathrm{d}{\mathit{\kappa}}_{\mathrm{1}}$.
This concludes the secondorder statistics in terms of the velocity that we consider in the main text to determine the mean energy dissipation rate.
In the following, we illustrate how one obtains an expression for the impact of randomsweeping effects on the dissipation rate estimate in terms of the turbulence intensity using the gradient method ${\mathit{\u03f5}}_{G}=\langle \mathit{\u03f5}\rangle [\mathrm{1}+\mathrm{5}{I}^{\mathrm{2}}]$ (Lumley, 1965; Wyngaard and Clifford, 1977). We consider a model wavenumber–frequency spectrum (Wilczek and Narita, 2012; Wilczek et al., 2014), which is based on the same modeling assumptions used in Wyngaard and Clifford (1977). It enables us to conduct a systematic assessment of the interplay between Taylor's hypothesis and the randomsweeping effects. The model wavenumber–frequency spectrum tensor Φ_{ij}(κ,ω) can be derived from an elementary linear randomadvection model (Kraichnan, 1964; Wilczek and Narita, 2012; Wilczek et al., 2014), which in the case of SHI turbulence can be expressed in terms of the energy spectrum tensor Φ_{ij}(κ):
Within the model, the wavenumber–frequency spectrum Φ_{ij}(κ,ω) consists of the energy spectrum tensor in wavenumber space Φ_{ij}(κ) multiplied by a Gaussian frequency distribution. Φ_{ij}(κ,ω) has a mean value determined by ω=κ_{1}U, i.e., Taylor's hypothesis expressed in Fourier space, and a variance proportional to the turbulence intensity. When the turbulence intensity tends toward zero at a fixed mean velocity, the frequency distribution tends toward a delta function, reestablishing the onetoone correspondence between the frequency and the wavenumber in the direction of the mean flow. To establish the connection to the different methods using longitudinal components and Taylor's hypothesis, we consider the $i=j=\mathrm{1}$ component of Eq. (B1). One obtains the estimate for the longitudinal wavenumber spectrum based on Taylor's hypothesis, which includes the effect of random sweeping, by first integrating over the wave vector space. This leads to the frequency spectrum, which corresponds to the one obtained from temporal singlepoint measurements of the longitudinal velocity component. Then, one applies Taylor's hypothesis, corresponding to the substitution ω=κ_{1}U, which leads to
Finally, this enables us to evaluate the influence of random sweeping on the gradient method since it is closely related to the wavenumber spectrum. Expressed in wavenumber space, the relation (Eq. 6) takes the following form, where we insert Eqs. (B1)–(B2) and solve the corresponding Gaussian integral over ω in the second step:
Hence, we recover, as expected, the result by Lumley (1965) and Wyngaard and Clifford (1977).
To numerically assess how random sweeping smears out the spectrum at finite turbulence intensities (see Fig. 4), we assumed a model wavenumber spectrum (Pope, 2000, Eq. 6.246 ff.):
where L is the energy injection scale and c_{L}=6.78, p_{0}=2 and β=2.094 are positive constants. Based on this model wavenumber spectrum, we first obtain E_{11}(κ) through Eq. (A7). One can then evaluate Eq. (B2), resulting in the spectrum being smeared out by random sweeping.
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 $\widehat{\mathit{n}}={\mathit{e}}_{\mathrm{3}}$ axis with the angle θ; see Fig. 5.
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 ${\mathit{e}}_{\mathrm{1}}^{\prime}$. The mean of this component gives us the estimate of the mean velocity of the probe U^{′}. Due to the misalignment between probe and true mean velocity, we underestimate the mean velocity ${U}^{\prime}=U\mathrm{cos}\mathit{\theta}$ compared to the true mean velocity U. As we apply Taylor's hypothesis to convert temporal increments into spatial distances, we therefore also underestimate the spatial distances:
Here $\stackrel{\mathrm{\u0303}}{r}$ denotes the estimated spatial distance and r the true distance that the probe moved.
The sampling direction of the probe, given by the direction of the mean velocity, e_{1}, can be expressed in the probe frame of reference:
Combining these two aspects, the vectorial distance covered by the probe in terms of the estimated one, therefore, becomes
In the probe frame of reference, the covariance tensor is given by (see Eq. A1)
By application of Taylor's hypothesis, we can evaluate this tensor at ${\mathit{r}}^{\prime}=\stackrel{\mathrm{\u0303}}{r}/\mathrm{cos}\mathit{\theta}(\mathrm{cos}\mathit{\theta}{\mathit{e}}_{\mathrm{1}}^{\prime}+\mathrm{sin}\mathit{\theta}{\mathit{e}}_{\mathrm{2}}^{\prime})$. So the corresponding longitudinal component $i=j=\mathrm{1}$ in terms of the estimated distance $\stackrel{\mathrm{\u0303}}{r}$ becomes
which, due to the misalignment, we interpret as the measured autocorrelation function. Then, the measured longitudinal integral length scale, Eq. (11), amounts to
where the integration of f(r) and g(r) is carried out in the last step (see Eq. 11) while considering the fact that ${L}_{\mathrm{22}}={L}_{\mathrm{11}}/\mathrm{2}$ for isotropic turbulence (Pope, 2000). Therefore, the energy injection scale estimate of the dissipation rate, Eq. (10), overestimates the dissipation rate due to misalignment as follows:
The same arguments that we applied to the covariance tensor, of course, also hold for the secondorder structure function tensor, Eq. (A2). Due to the effects of misalignment, it takes the form
where in the inertial range of SHI turbulence the transverse secondorder structure function ${D}_{\mathrm{NN}}\left(r\right)={D}_{\mathrm{22}}\left(r\right)={D}_{\mathrm{33}}\left(r\right)$ can be expressed as ${D}_{\mathrm{NN}}\left(r\right)=\mathrm{4}{D}_{\mathrm{LL}}\left(r\right)/\mathrm{3}=\mathrm{4}{C}_{\mathrm{2}}(r\mathit{\u03f5}{)}^{\mathrm{2}/\mathrm{3}}/\mathrm{3}$ (Pope, 2000). Therefore, the estimated dissipation rate using the secondorder SF method (cf. Eq. 7) is affected by misalignment as
The misalignment error for the gradient method can be estimated analytically starting from the longitudinal component of the velocity gradient covariance tensor ${R}_{\mathrm{1111}}^{\prime}$, Eq. (A4). Following similar arguments as above and starting from Eq. (A4), we obtain
where we expressed the estimated distance through the true one $\stackrel{\mathrm{\u0303}}{r}=r\mathrm{cos}\mathit{\theta}$ in the second line. Using ${\partial}_{r}^{\mathrm{2}}g\left(r\right)=\mathrm{2}{\partial}_{r}^{\mathrm{2}}f\left(r\right)+\frac{r}{\mathrm{2}}{\partial}_{r}^{\mathrm{3}}f\left(r\right)$ (Pope, 2000), the velocity gradient covariance tensor reduces to
where $\langle {{u}^{\prime}}^{\mathrm{2}}\rangle {lim}_{r\to \mathrm{0}}{\partial}_{r}^{\mathrm{2}}f\left(r\right)=\langle (\partial u/\partial {x}_{\mathrm{1}}{)}^{\mathrm{2}}\rangle $ (Pope, 2000) is used for the last step. The energy dissipation rate based on the gradient method, Eq. (6), is overestimated by the very same factor due to misalignment:
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 HighPerformance 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 KonradAdenauerStiftung. Tobias Bätge was financially supported by a fellowship of the IMPRS for Physics of Biological and Complex Systems.
The article processing charges for this openaccess publication were covered by the Max Planck Society.
This paper was edited by Luca Mortarini and reviewed by two anonymous referees.
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 Abstract
 Introduction
 Methods
 Results and discussion
 Practical guidelines
 Summary
 Appendix A: Preliminaries on secondorder statistics
 Appendix B: Impact of randomsweeping effects on the gradient method
 Appendix C: Effect of probe misalignment
 Code availability
 Data availability
 Author contributions
 Competing interests
 Disclaimer
 Acknowledgements
 Financial support
 Review statement
 References
 Abstract
 Introduction
 Methods
 Results and discussion
 Practical guidelines
 Summary
 Appendix A: Preliminaries on secondorder statistics
 Appendix B: Impact of randomsweeping effects on the gradient method
 Appendix C: Effect of probe misalignment
 Code availability
 Data availability
 Author contributions
 Competing interests
 Disclaimer
 Acknowledgements
 Financial support
 Review statement
 References