2.1. Description of the database

The computations giving the two datasets (I and II) are carried out for three-dimensional flows, performed in $512 \times 512 \times 512$ and $1024 \times 1024 \times 1024$ systems of grid points for extended time periods of 1167 and 1841 time steps, respectively. The box sizes are $2{\rm \pi} \times 2{\rm \pi} \times 2{\rm \pi}$ in computational units for both cases. The system is periodic in all directions. From the numerically obtained flow field, a large number ($1.92 \times 10^5$ and $3.64 \times 10^5$, respectively) of point-particle trajectories are constructed and these form the basis for the following analysis. (It is not practically feasible to store the entire space–time information of the velocity field $\boldsymbol {u}(\boldsymbol {r},t)$.) The database can be seen as a numerical equivalent of, for instance, the laboratory experiment of Ott & Mann (Reference Ott and Mann2000) and Mann et al. (Reference Mann, Ott, Pécseli and Trulsen2005) where the trajectories of a large number of small polystyrene spheres were followed by image velocimetry techniques. The basic parameters of the two numerical simulations are summarized in table 1, where $Re_{\lambda }$ denotes the microscale Reynolds number and the Kolmogorov microscale is $\eta _K \equiv (\nu ^3/\epsilon )^{1/4}$, while $\tau _K \equiv (\nu /\epsilon )^{1/2}$ is the Kolmogorov time scale. Parameters are given in standard notation (Biferale et al. Reference Biferale, Boffetta, Celani, Lanotte and Toschi2005b) using computational units. We have $\epsilon$ to be approximately the same for both simulations, but the specific viscosities $\nu$ differ. Consequently the two Reynolds numbers as well as the two Kolmogorov scales are substantially different. The universal subranges are different for the two cases. The uncertainties in the results are confined to regions for small particle separations, and the estimated error is illustrated by grey shading in figure 1.

Figure 1.

The numerically obtained normalized longitudinal structure function $\mathcal {S}(r)/u_K^2\equiv \langle (u_\parallel ({\boldsymbol \xi },t) - u_\parallel ({\boldsymbol \xi }+\boldsymbol {r},t))^2\rangle /u_K^2$, shown on a double logarithmic scale for varying normalized separation $r/\eta _K$, with $\eta _K = (\nu ^3/\epsilon )^{1/4}$ being the Kolmogorov length scale (see also Pécseli et al. Reference Pécseli, Trulsen, Stiansen and Sundby2020). The Kolmogorov velocity $u_K = (\nu \epsilon )^{1/4}$ is used for normalization. Results from the two simulations (I and II) are shown with blue dashed and black solid lines, respectively. Analytical results are shown for the inertial and viscous subranges by the slopes of the red lines with $r^{2/3}$ and $r^2$, respectively. For clarity of presentation, the two slopes as well as an empirical analytical approximation (thin red line) have been offset vertically by a factor of 2. The vertical arrow indicates the ‘cross-over’ length between the two subranges at $r = \eta _0$. The uncertainty at small separations is indicated by grey shading. This shading is representative also for the following figures 2, 3, 8 and 9.

Table 1.

Simulation I lasts 1167 time steps with $\textrm {d}t = 0.005$; simulation II lasts 1841 time steps with $\textrm {d}t = 0.0023$. All values in computational units. The time duration of the simulations is $T$. The value of $u_{rms}$ given is an average of the root-mean-square values of the three velocity components.

Since only simulation particle trajectories are available, the numerical results cannot be used for directly obtaining any frequency spectrum from single-point measurements. By using two-point measurements of velocity components of many pairs of particles, $u_j({\boldsymbol \xi },t)$ and $u_j({\boldsymbol \xi }+\boldsymbol {r},t)$ taken at the same time $t$ and different $\boldsymbol {r}$, a structure function can, however, be determined just as in the laboratory experiment of Ott & Mann (Reference Ott and Mann2000). The analysis can be extended to points also at different times, i.e. $u_j({\boldsymbol \xi },\vartheta )$ together with $u_j({\boldsymbol \xi }+\boldsymbol {r},\vartheta + t)$, to obtain estimates for the full space–time-varying structure function. The two particles are selected with the only constraints being on their space–time separation, irrespective of their ‘history’. The space–time-varying structure function can be used for tests of the Taylor hypothesis, as discussed in the following. The analysis has a Lagrangian counterpart which is discussed separately.

2.2. Basic results

To demonstrate the quality of the data used for the present study, we first show the structure function $\mathcal {S} (r,t) \equiv \langle (u_\parallel ({\boldsymbol \xi },\vartheta ) - u_\parallel ({\boldsymbol \xi }+\boldsymbol {r},\vartheta + t))^2\rangle$, with $u_\parallel$ being the velocity component along the separation vector $\boldsymbol {r}$. In the following we use the abbreviation $\mathcal {S}(r)$ for $\mathcal {S}(r,t=0)$. Using the basic parameters $\epsilon$ and $\nu$, a well-known argument (Chandrasekhar Reference Chandrasekhar1957) gives the only dimensionally correct combination (Buckingham Reference Buckingham1914) to be

(2.1)\begin{equation} \mathcal{S} (r)= \langle(u_\parallel ({\boldsymbol \xi},t) - u_\parallel ({\boldsymbol \xi}+\boldsymbol{r},t))^2\rangle = C_K (\epsilon \nu)^{1/2} F(r\epsilon^{1/4}/\nu^{3/4}) \end{equation}

for homogeneous and isotropic conditions, where $F(z)$ is a dimensionless function of a dimensionless variable $z$ and $C_K$ is the Kolmogorov constant. In an inertial subrange, independent of $\nu$, we require $F(z) = z^{2/3}$ in order make the viscosity $\nu$ vanish from the expression (Chandrasekhar Reference Chandrasekhar1957; Beran Reference Beran1968), thereby giving the well-known result for the structure function in the inertial subrange, $C_K (\epsilon r)^{2/3}$. The Kolmogorov constant is determined empirically to be in the range $C_K\in \{2.1, 2.5\}$. The same dimensional arguments (Chandrasekhar Reference Chandrasekhar1957) will give the expression for the temporal structure function of any velocity component to be

(2.2)\begin{equation} \langle(u ({\boldsymbol \xi},\vartheta) - u ({\boldsymbol \xi},\vartheta + t))^2\rangle = C_T \epsilon t \end{equation}

in a subrange independent of $\nu$, with $C_T$ being a universal dimensionless constant (Du et al. Reference Du, Sawford, Wilson and Wilson1995; Falkovich et al. Reference Falkovich, Xu, Pumir, Bodenschatz, Biferale, Boffetta, Lanotte and Toschi2012). The result (2.2) is ambiguous in one respect: the result can be obtained experimentally either by sampling the flow in a fixed position as a function of time or by following an ensemble of particles along their Lagrangian orbits to obtain the time-varying velocity (Sawford & Yeung Reference Sawford and Yeung2015). In the first case, the time variation will in part be due to smaller-scale universal eddies being swept past the observer by large-scale non-universal eddies, and the result thus be ‘contaminated’ by non-universal effects (Tennekes & Lumley Reference Tennekes and Lumley1972; Tennekes Reference Tennekes1975). Using a second-order Lagrangian model of grid turbulence the Kolmogorov constant $C_T$ was estimated by Du et al. (Reference Du, Sawford, Wilson and Wilson1995) for the Lagrangian structure function, giving $C_T\in \{3.0\pm 0.5\}$.

Analytical results for the structure function in the viscous subrange (Davidson Reference Davidson2004) give $\langle (u_\parallel ({\boldsymbol \xi },t) - u_\parallel ({\boldsymbol \xi }+\boldsymbol {r},t))^2\rangle = C_\nu r^2 \epsilon /\nu$. The numerical coefficient is here obtained analytically as $C_\nu = 1/15$. The separation between the inertial and viscous subranges is found for $C_K (\epsilon r_0)^{2/3} = C_\nu r_0^2 \epsilon /\nu$, i.e. at $r_0 = (15 C_K)^{3/4}\eta _K \approx 13 \eta _K\equiv \eta _0$. The difference between $\eta _K$ and $\eta _0$ is not trivial (Pécseli & Trulsen Reference Pécseli and Trulsen2007). For ideally frozen turbulence this length scale translates to a time $\eta _0/V$. In most laboratory experiments, the Kolmogorov scale is in or below the millimetre range and the viscous subrange cannot be resolved. For conditions met in nature, however, also this subrange has importance (Kiørboe Reference Kiørboe2008), so for completeness we show representative data relevant there in table 2.

Table 2.

The specific energy dissipation $\epsilon$ (dissipated energy per unit mass of fluid) and other characteristic turbulence parameters as found in the oceans (Granata & Dickey Reference Granata and Dickey1991; Kiørboe & Saiz Reference Kiørboe and Saiz1995; Pécseli et al. Reference Pécseli, Trulsen, Stiansen and Sundby2020).

Approximations for the structure functions are found for the combined intermediate separations $r$, i.e. the inertial subrange, and for small $r$, i.e. the viscous subrange (see Appendix B). In Appendix A we discuss in detail a model for the space–time variation of the structure function in the inertial subrange. For large $r$ outside the inertial range, the structure function will approach the value $2 u_{rms}^2$, with $u_{rms}$ again being the root-mean-square value of the velocity component chosen for the structure function.

Normalized numerical results for the structure functions are shown in figure 1 for the two datasets. The results have been plotted for separations up to $r \approx {\rm \pi}$ in computational units. The figure uses double logarithmic axes and the normalizations are made by the length $\eta _K$ and velocity $u_K$ scales. In this form we find an excellent agreement between the two simulations. Earlier results (Pécseli et al. Reference Pécseli, Trulsen, Stiansen and Sundby2020) used a reduced sampling rate during the evaluation of the structure function. This has effect for the statistical uncertainty of the results for small $r$. The analytical approximation discussed in Appendix B is very good and agrees with the numerical results within the line thickness in the plot. We find this observation to be worth emphasizing. Figure 1 is our reference, i.e. the one that should be reproduced by use of Taylor's hypothesis, should it be ideally applicable.

Also the transverse structure functions $\mathcal {G} (r)\equiv \langle (u_\perp ({\boldsymbol \xi },t) - u_\perp ({\boldsymbol \xi }+\boldsymbol {r}, t))^2\rangle$ were determined with $u_\perp$ being either one of the velocity components $\perp \boldsymbol {r}$. Apart from a difference in the numerical factor, also $\mathcal {G} (r)$ follows the universal $(\epsilon r)^{2/3}$ scaling, indicating that the smallest scales have reached an isotropic statistical equilibrium. A large-scale eddy turnover time can be defined as $T_t\equiv L/u_{rms}$. With $L\approx {\rm \pi}$ in computational units we find $T_t\approx 2$ for both simulations. This time is of the same order of magnitude as the simulation time $T$, see table 1, so the largest eddies may not have reached an equilibrium. It is found that the root-mean-square values of the three velocity components are somewhat different for the two datasets. We find the normalized root-mean-square values of the three components $u_{rms,j}/u_K$ with $j = 1, 2, 3$ to be $\{7.7, 6.3, 6.7\}$ and $\{10.6, 7.2, 8.1\}$ for datasets I and II, respectively.

Both the viscous and inertial subranges of the turbulence are well resolved for both simulations. There are no indications of a ‘bottleneck’ effect (Dobler et al. Reference Dobler, Haugen, Yousef and Brandenburg2003) in the power spectrum obtained by Fourier transform of the autocorrelation function entering $\mathcal {S} (r)$. Physically, the bottleneck effect arises because of the finite resolution of the sub-Kolmogorov scales in numerical simulations, giving a lack of small-scale vortices. This makes the energy cascade less effective around the Kolmogorov length scale, as compared with the ideal physical conditions. Some numerical results seem to indicate that the bottleneck effect is a consequence of viscous effects stabilizing small vortex tubes against the kink instability (Woodward et al. Reference Woodward, Porter, Edgar, Anderson and Bassett1995). These problems do not arise here. We conclude that the quality of the database is such that we can test the validity of also the Taylor hypothesis for both inertial and viscous subranges with confidence by using these data.

2.3. Space–time-varying structure function

To test the Taylor hypothesis we choose the space–time-varying longitudinal structure function $\mathcal {S} (r, t)\equiv \langle (u_\parallel ({\boldsymbol \xi },\zeta ) - u_\parallel ({\boldsymbol \xi }+\boldsymbol {r},\zeta + t))^2\rangle$. The particle-trajectory data are searched for particles at some position and then at a later time $t$ at a displaced position $\boldsymbol {r}$ within a narrow $|r|$ interval. The velocity differences are recorded and the data subsequently averaged. The time resolution is given by the computational time step. Similarly, also the space–time-varying transverse structure function $\mathcal {G} (r, t)$ can be obtained.

In figures 2 and 3 we show numerically obtained longitudinal structure functions for the two datasets. To cover an $r$ range extending by more than four orders of magnitude, the samples for the evaluation of the Eulerian structure function are collected in bins with size varying exponentially with $r$. At small $r$ the bin size is changed to a linear variation in order to balance the requirements of both spatial resolution and statistical uncertainty. For the range $r/\eta _K < 3$ the number of samples per bin varies like $r^2$ and more rapidly for larger $r$ values. This gives rise to an increasing statistical uncertainty by several orders of magnitude towards the smallest $r$ values. For this reason a least squares local second-order polynomial approximation noise reduction has been applied for $r/\eta _K < 3$ in figures 2 and 3 as well as in figure 1. For the three orders of magnitude, extended discrete $t$-range similar measures are incorporated. Reference dashed lines are inserted for normalized velocities $V/u_{rms} = \{0.1, 0.3, 1, 3, 10, 100\}$. This normalization turns out to be the most suitable for the following. To obtain the limiting case in figure 1 we have to sample along a line with $V/u_{rms}\rightarrow \infty$.

Figure 2.

Estimate for the normalized longitudinal structure function $\mathcal {S} (r,t)/u_K^2$ using dataset I, shown as a function of normalized variables $r/\eta _K$ and $t/\tau _K$ with logarithmic scales on all axes. Solid black lines are for the levels $\{0.1, 0.3, 1, 3, 10, 30\}$, the dashed lines for normalized velocities $V/u_{rms} = \{0.1, 0.3, 1, 3, 10, 100\}$. These lines will be parallel when projected on the $(r,t)$ plane for logarithmic scales. The time separation axis begins at $t = 0.02 \tau$, the spatial separations at $r = 0.4 \eta _K$. Thin black parallel lines are given for selected $t$ values and serve only as a guide for the eye.

Figure 3.

Estimate for the normalized longitudinal structure function using dataset II, shown as a function of normalized variables $r/\eta _K$ and $t/\tau _K$ with logarithmic scales on all axes. Levels and normalizations are as in figure 2.

In the viscous and inertial subranges, the spatially varying structure function $\mathcal {S}(r)$ is insensitive for the large-scale dynamics and the value of $u_{rms}$, except for a transition zone between the inertial and non-universal ranges. Its time-varying Eulerian counterpart $\mathcal {S}(t)$ will, on the other hand, contain significant contributions from the large-scale energy-containing non-universal eddies (Tennekes Reference Tennekes1975) characterized by velocities of the order of $u_{rms}$. A transition region in $\mathcal {S}(r,t)$ for varying $(r,t)$ can thus be expected, but the details of this transition have not been discussed in the literature. To gain more information, we suggest taking the ratio of the total and the longitudinal structure functions $\mathcal {S}_T(r,t) \equiv \langle (\boldsymbol {u}({\boldsymbol \xi },\vartheta )-\boldsymbol {u}({\boldsymbol \xi }+\boldsymbol {r},\vartheta + t))^2\rangle \equiv 2\mathcal {G}({r},t) + \mathcal {S}({r},t)$ and $\mathcal {S}({r},t)$. The result is shown in figure 4. For vanishing time separations both have the same universal scalings with $r$ and $\epsilon$ in the viscous and inertial subranges, with only the numerical coefficients being different for the longitudinal and transverse cases. For large time separations the large non-universal eddies will deteriorate the proportionality of the two structure functions. For $r\rightarrow 0$ the difference between the transverse $\mathcal {G}$ and longitudinal $\mathcal {S}$ structure functions vanishes, and the ratio $(2 \mathcal {G} + \mathcal {S})/\mathcal {S} \rightarrow 3$ for all times $t > 0$.

Figure 4.

Colourplot of the ratio $\mathcal {S}_T(r,t)/\mathcal {S}(r,t)$ for dataset II plotted on double logarithmic axes. The construction of the figure uses the longitudinal as well as both transverse velocity components. The solid black line is drawn for the level 11/3. Some noise is found in the data for small spatial separations, for reasons already mentioned.

To get a numerical reference value for comparison with the ratio used for illustration in figure 4 we have for vanishing time separations the general relation (Batchelor Reference Batchelor1953) $\mathcal {G}(r) = \mathcal {S}(r)+ \frac {1}{2} r \mathcal {S}^\prime (r)$. For any power-law relation $\mathcal {S}(r) = A r^\alpha$ we have $\mathcal {G}(r)/\mathcal {S}(r) = 1 + \frac {1}{2} \alpha$ applicable for both the viscous and the inertial subranges. For the inertial subrange we find $\mathcal {G}(r)/\mathcal {S}(r) = 4/3$ while the viscous subrange gives $\mathcal {G}(r)/\mathcal {S}(r) = 2$. The transition from the viscous to the inertial subrange is here found at a separation length scale of approximately $10\eta _K$ for the transverse structure function, to be compared with the separation for the parallel structure function at approximately $13 \eta _K$. In figure 4 the results should be compared to values 11/3 and 5 for the two subranges also for $t > 0$. Some details are seen better in figure 5. For $t = 0$ we note a clear separation between the viscous and inertial subranges along the $r/\eta _K$ axis, and also a well-defined transition region in the $(r,t)$ plane as the time separation is increased. This transition is not along a straight line in the $(r,t)$ plane, although it can be approximated by $r = t u_{rms}$ for $r > \eta _0$. The grey area in figure 4 corresponds to a level close to the value 3. The results in figures 4 and 5 invite the interpretation that the velocity vector directions (parallel and perpendicular) to a separation vector, for $r > \eta _0$, become decoupled for times larger than $r/u_{rms}$, i.e. $\mathcal {G}(r > \eta _0,t > r/u_{rms}) \approx \mathcal {S}(r > \eta _0,t > r/u_{rms})$. Similar observations can be made for $r$ in the viscous subrange, but here the separation line does not have a correspondingly simple form, although $r = t u_{rms}$ can be an approximation also here. If the two points with separation $r$ are selected at the same time with separations in the universal subrange, they will move without relative displacement when advected by eddies with scale sizes larger than the separation (Csanady Reference Csanady1973; Mikkelsen, Larsen & Pécseli Reference Mikkelsen, Larsen and Pécseli1987; Del Álamo & Jiménez Reference Del Álamo and Jiménez2009). The separation between the two points is on the other hand controlled by eddies smaller than the separation distance. In this limit the root-mean-square value of the velocity is immaterial. If, however, the two points are sampled at different times, the separation of the points is influenced also by the large-scale eddy motion. The larger the time separation, the stronger is the effect of large scales. With a spatial separation $r$ we associate eddies with a similar scale. A corresponding eddy turnover time, or coherence time, is $\tau _r \approx \epsilon ^{-1/3} r^{2/3}$ (Tennekes Reference Tennekes1975), while a characteristic velocity is $u_r \approx (\epsilon r)^{1/3}$. Due to the motion of the large-scale non-universal eddies, the smaller inertial eddies are displaced a distance $\ell _e\approx u_{rms}/t$ in the time interval $t$ entering the structure functions. When $\ell _e > r$ and $t > \tau _r$ the correlation is reduced between the directions of the two samples of the local fluid velocities entering the structure function. Scales larger than $r$ retain, on the other hand, their coherence at the same time separation. Appendix A contains more details of the arguments, showing also an illustrative model. Scale-dependent advection velocities are also discussed here.

Figure 5.

The ratio $\mathcal {S}_T(r,t)/\mathcal {S}(r,t)$ derived from the longitudinal and transverse structure functions for dataset II is shown here on a linear vertical scale for selected times $t/\tau _K = 0.0$, 0.04, 0.35, 0.70, 2.51, 6.84 and 17.45. For $r > \eta _0$ we find a plateau developing near 11/3 shown by a dashed line, while for $r\rightarrow 0$ with $t = 0$ we find a saturation at the level 5. The curves are uncertain for $r < \eta _K$ because only few samples for the estimation of the structure functions are found at these small separations. The noise for small $r$ seen in figure 4 is found also here.

2.4. The Lagrangian structure function

For comparison with the analytical result (2.2) we show in figure 6 the normalized Lagrangian structure function $\mathcal {S}_{Lx}(t)/u_K^2 \equiv \langle (u_x ({\boldsymbol \xi }(\vartheta ),\vartheta ) - u_x ({\boldsymbol \xi }(\vartheta + t),\vartheta + t))^2\rangle /u_K^2$ as a function of normalized time (Biferale et al. Reference Biferale, Bodenschatz, Cencini, Lanotte, Ouellette, Toschi and Xu2008; Sawford & Yeung Reference Sawford and Yeung2015). The initial variation follows a clear $t^2$ variation, which becomes closer to a $t$-proportional variation at later times, $t/\tau _K > 2$. The agreement is not perfect, indicating that a significant fraction of the Lagrangian trajectories entering the averaging are short and within the viscous subrange, while others are long and close to the outer scales in the non-universal part of the turbulence. In figure 7 we show (on double logarithmic axes) samples from the distribution of the lengths of trajectories entering the construction of the Lagrangian structure function, illustrating the systematic increase in average trajectory lengths with increasing time. The separation between the inner and outer scales in the present simulations is not sufficient to give the analytical result (2.2) unambiguously. The initial $t^2$ variation is explained by a short time series expansion of $u_x({\boldsymbol r}(t),t) \approx u_x(0,0) + \textrm {d}u_x/\textrm {d}t|_{t=0}\, t$ in the Lagrangian structure function. For large time separations $t/\tau _K$ the velocities $u_x ({\boldsymbol \xi }(\vartheta ),\vartheta )$ and $u_x ({\boldsymbol \xi }(\vartheta + t),\vartheta + t)$ become independent and $\mathcal {S} \rightarrow 2\langle u_x^2\rangle$ for the Lagrangian structure function. This limit is not accessible within the spatial and temporal ranges available in the simulations, but in figure 6 we note that the curve begins to ‘flatten out’ for large $t/\tau _K$. Indications are that the Lagrangian time scale is shorter than its Eulerian counterpart, in agreement with some analytical results (Weinstock Reference Weinstock1976). The difference between the time exponents for the Lagrangian structure function and its Eulerian counterpart taken for vanishing spatial separations demonstrates the limitations of the result in (2.2) as discussed before. The time variation of the Eulerian structure function in the inertial subrange is very different from the corresponding variation of its Lagrangian counterpart (Tennekes & Lumley Reference Tennekes and Lumley1972).

Figure 6.

Estimate for the normalized Lagrangian structure function $\mathcal {S}_{Lx}(t)/u_K^2$ using dataset II, shown as a function of normalized time variable $t/\tau _K$ with logarithmic scales on both axes. Solid red reference lines give the slopes for $t^2$ and $t$. The thin red curve gives the empirical transition between the viscous and the inertial ranges corresponding to the transition parameter $p = 5/2$ (see Appendix B). The underlying green curve illustrates the corresponding analytical model for the transition from the universal range to the asymptotic limit for a transition parameter $p = 2$.

Figure 7.

Snapshots of the normalized distribution function $f_r$ of normalized lengths $r/\eta _K$ measured from start-point to end-point of trajectories that enter the construction of the Lagrangian structure function in figure 6. Selected snapshot times are $t/\tau _K \approx 1, 3, 10, 20$ and 35. The figure uses double logarithmic axes.

2.5. Implementation of Taylor's hypothesis

The velocity fluctuations in a turbulent environment in nature can be sampled by a detector moving with velocity $\boldsymbol {V}$ along a straight-line trajectory at two times $\vartheta$ and $\vartheta + t$, corresponding to two positions ${\boldsymbol \xi }$ and ${\boldsymbol \xi } + \boldsymbol {r}\equiv {\boldsymbol \xi } + \boldsymbol {V} t$. Repeating the measurement many times, an ensemble of realizations can be obtained and the average of the mean square velocity difference can subsequently be estimated. The result will correspond to a point on the structure function $\mathcal {S} (r = V t, t)$. With a spatial correlation length $r_c$, the range of relevant time variation is $t \in \{0,r_c/V\}$, i.e. for times outside this range, the correlation function is negligible. When $V$ is large, the range of time variation is small and we have $\mathcal {S}(r = V t, t)\approx \mathcal {S} (r = V t, 0) \equiv \mathcal {S} (r=V t)$ which represents the Taylor hypothesis applied for the structure function. The question is how large $V$ has to be for the approximation to be valid.

In figures 8 and 9 we show results for structure functions $\mathcal {S}(r,r/V)$. The data are obtained by sampling the structure function $\mathcal {S}(r,t)$ in figure 2 along lines $r = V t$ for $V/u_{rms} = 0.1, 0.3, 1, 3, 10$ and $100$. The red lines give the analytical approximation (B1) for $\mathcal {S} (r)$. The results for $V/u_{rms} = 3, 10$ and $100$ are hardly distinguishable and agree entirely with $\mathcal {S} (r)$ within the inertial subrange.

Figure 8.

Sampling of the normalized structure function $\mathcal {S}$ along lines $r = V t$ for $V/u_{rms} = 0.1$, 0.3, 1, 3, 10 and 100 from figure 2 (dataset I). The thin red line gives the analytical approximation (B1) for $\mathcal {S} (r)$. The results for $V/u_{rms} = 3$, 10 and 100 are hardly distinguishable with the given line thickness. The top curve is for $V/u_{rms} = 0.1$ with $V$ increasing from top to bottom.

Figure 9.

Results corresponding to figure 8, here using figure 3 (dataset II).

As $V$ is decreased we observe several features in figures 8 and 9. (i) Deviations from the ideal results $\mathcal {S} (r)$ are first found for the viscous subrange for decreasing $V$. The power law (i.e. the slope in a double logarithmic presentation) for the viscous subrange remains approximately constant when reducing $V$. (ii) The sampled curves move consistently to higher values of $\mathcal {S}/u_K^2$ as $V$ is reduced. A power law for the structure function remains an acceptable approximation, but the exponent reduces for decreasing $V$. (iii) The separation point between viscous and inertial subranges moves towards smaller $r/\eta _K$ for decreasing $V$ when $V < u_{rms}$. (iv) The estimates of $\epsilon$ are steadily increasing for decreasing sampling velocities $V$. Considering the inertial subrange only, we find $\epsilon$ to be determined with an accuracy of 10 % for $V/u_{rms} = 3$, and of approximately 25 % for $V/u_{rms} = 1$. Even for a value as low as $V/u_{rms} = 0.3$ we find the difference between the ideal and the sampled result in the inertial subrange to agree within a factor of 3, noting though that the slope of the sampled curve deviates from the analytical value $2/3$. As a ‘rule of thumb’ it can be argued that when $V/u_{rms} \geq 0.7$ we have $\epsilon$ determined with an accuracy comparable to or better than that found for $C_0$. An order-of-magnitude (or better) estimate for $\epsilon$ can be obtained for $V/u_{rms} \geq 0.3$.

In figures 10 and 11 we show the variation of $\mathcal {S} (0, t)$. Since the exact value $r = 0$ is not obtainable in the analysis, we use the three smallest available $r$ separations: $r/\eta _K = 0.6, 0.8$ and 1.0 for dataset I and $r/\eta _K = 0.4, 0.6$ and 0.8 for dataset II. The analytical form (2.2) finds no support in figure 10, for reasons already mentioned. Alternatively it was hypothesized by Tennekes (Reference Tennekes1975) that a fixed observer does not experience the $(\epsilon t)$ scaling of the structure function as found by dimensional reasoning, but rather a time variation caused by the smaller inertial eddies swept past the observer by the large-scale non-universal part of the spectrum, where most of the energy is found (see also Appendix A). By this argument we expect that $(\epsilon r)^{2/3} \rightarrow ( \epsilon t u_{rms} )^{2/3}$, where it is implicitly assumed that the large non-universal eddies contain nearly all of the energy. A more accurate estimate is found by taking a plausible velocity probability density to be a Maxwellian giving $\langle u_\parallel ^{2/3}\rangle = u_{rms}^{2/3} \times 2^{1/3} \varGamma (5/6) / \sqrt {{\rm \pi} } \approx 0.80 u_{rms}^{2/3}$ to replace $u_{rms}$ in the foregoing expression.

Figure 10.

Estimate of the time variation of the normalized Eulerian structure function $\mathcal {S} (0, t)$ for dataset I. The exact value $r = 0$ is not available due to the construction of the present database, so we show results for the three smallest available $r$ separations for dataset I, $r/\eta _K = 0.6, 0.8$ and 1.0; the smallest separation is at the bottom. The lines are slightly irregular due to the limited number of samples for small $r$ separations. Two lines for $t^{2}$ and $t^{1/2}$ are inserted in red for reference. On average, the smallest $r$ separation (steepest $t$ variation, shown with black line) comes closest to the result $\sim t^2$ for $t/\tau _K < 1$. The thin red line represents the $p = 1$ fit discussed in Appendix B. Note that power-law approximations become inaccurate for $t > \tau _K$.

Figure 11.

Results corresponding to those shown in figure 10, here using dataset II. The black, green and blue curves correspond to $r/\eta _K = 0.4, 0.6$ and 0.8.

The normalized separation time scale between the two subranges is estimated to be $t_0/\tau _K = 1.5\pm 0.5$ for the two datasets taken together. For comparison we have the result obtained by the suggestion of Tennekes (Reference Tennekes1975) giving $t_0/\tau _K = \eta _0/( u_{rms} \tau _K) = 1.79$ using the values from table 1. The agreement is reasonable considering the uncertainties involved. Since the scales in the viscous subrange contain the smallest part of the energy, the scale-size dependence of the advection velocities (Del Álamo & Jiménez Reference Del Álamo and Jiménez2009; Renard & Deck Reference Renard and Deck2015) is of minor importance there. Consequently, the argument of Tennekes (Reference Tennekes1975) applies best for this subrange.

We tested the estimate $( \epsilon t u_{rms} )^{2/3}$ argued before for the time variation of the fixed-point Eulerian structure function (see also Appendix A). It seems that the result has little support in our analysis. This observation has support in the analysis of Wyngaard & Clifford (Reference Wyngaard and Clifford1977). The agreement was worst for dataset II. We find it unlikely that the Eulerian fixed-point results can be used for estimating $\epsilon$, but studies for very high Reynolds numbers might give more promising results.