Direct assessment of Kolmogorov's first refined similarity hypothesis

TL;DR

This study provides the first conclusive experimental and numerical evidence supporting Kolmogorov's first refined similarity hypothesis, demonstrating the universality of local velocity increment distributions across scales in turbulent flows.

Contribution

It offers the first direct experimental and numerical validation of Kolmogorov's first refined similarity hypothesis using three-dimensional velocity measurements.

Findings

Distributions of velocity increments collapse when scaled by local Reynolds number.

Supports universality of the refined similarity hypothesis.

Provides conclusive experimental evidence for the hypothesis.

Abstract

Using volumetric velocity data from a turbulent laboratory water flow and numerical simulations of homogeneous, isotropic turbulence, we present a direct experimental and numerical assessment of Kolmogorov's first refined similarity hypothesis based on three-dimensional measurements of the local energy dissipation rate $\epsilon_r$ measured at dissipative scales $r$. We focus on the properties of the stochastic variables $V_L = \Delta u(r)/(r \epsilon_r)^{1/3}$ and $V_T = \Delta v(r)/(r\epsilon_r)^{1/3}$, where $\Delta u(r)$ and $\Delta v(r)$ are longitudinal and transverse velocity increments. Over one order of magnitude of scales $r$ within the dissipative range, the distributions of $V_L$ and $V_T$ from both experiment and simulation collapse when parameterised by a suitably defined local Reynolds number, providing the first conclusive experimental evidence in support of the first refined similarity hypothesis and its universality.

Equations

$${ϵ}_r(\mathbf{X},r,t)=\frac{6}{\pi r^3}{\iiint }_{|\mathbf{y}|\le r/2}\mathrm{d}\mathbf{y}ϵ(\mathbf{X}+\mathbf{y},t)$$

$$\displaystyle\langle r|\Delta u|/\nu\big{|}Re_{r}\rangle$$

$$\displaystyle\langle r|\Delta v|/\nu\big{|}Re_{r}\rangle$$

Full text

Obtaining a universal statistical description of hydrodynamic turbulence has been a widely-pursued yet elusive objective within fluid mechanics. Kolmogorov’s refined similarity hypotheses represent one such seminal attempt (Kolmogorov, 1962), which underpins the modern understanding of intermittency in small scale turbulence (Sreenivasan and Antonia, 1997). This phenomenon directly influences, amongst others, the efficiency of rain formation in clouds (Shaw, 2003), the production of pollutants in combustion processes (Sreenivasan, 2004) and the propagation of sound and light through the atmosphere (Tatarskii, 1971; Wilson et al., 1996). In this Rapid Communication, we overcome previous technical limitations to provide a quantitative and direct experimental assessment of the validity of the first refined similarity hypothesis with back-to-back comparisons against numerical simulations to examine their universality.

The similarity hypotheses describe turbulent flows in terms of velocity differences, or increments, $\Delta\bm{u}=\bm{u}(\bm{x},t)-\bm{u}(\bm{x}^{\prime},t)$ between simultaneously measured pairs of points in the flow, where the spatial separation $\bm{r}=\bm{x}-\bm{x}^{\prime}$ is much smaller than the energy injection scale $L$. In their simplest formulation, known as K41 (Kolmogorov, 1941), the distribution of velocity increments is prescribed by the scale $r=|\bm{r}|$, the average rate of kinetic energy dissipation $\langle\epsilon\rangle$ and the fluid kinematic viscosity $\nu$. Laboratory and numerical experiments now widely confirm departures from the K41 scaling (Anselmet et al., 1984; Nelkin, 1994; Sreenivasan and Antonia, 1997). The essence of this deviation was first articulated by Landau (Landau, 1987), who remarked that whilst the increment distribution may plausibly uniquely depend upon a temporally localised average of the energy dissipation rate $\epsilon(\bm{x},t)=\nu\Sigma_{i,j}(\partial u_{i}/\partial x_{j}+\partial u_{j}/\partial x_{i})^{2}/2$, the distribution law of $\Delta\bm{u}$ must depend upon the fluctuation of this local average over time, which may in turn depend upon the whims and fancies of the largest scale motions that feed the turbulence its energy.

Landau’s criticisms are accounted for in the refined similarity scaling (Kolmogorov, 1962; Oboukhov, 1962), known as K62, by substituting $\langle\epsilon\rangle$ for a local dissipation rate $\epsilon_{r}$

[TABLE]

which is a spatial average of the instantaneous energy dissipation field over a sphere whose poles are defined by $\bm{x}$ and $\bm{x}^{\prime}$, centered at $\bm{X}=(\bm{x}+\bm{x}^{\prime})/2$ with diameter $r$ 111This definition is consistent with Oboukhov’s formulation; in Kolmogorov’s formulation, the averaging volume is a sphere of radius $r$ centred at $\bm{x}$.. This permits a characteristic velocity scale $U_{r}\equiv(r{\epsilon}_{r})^{1/3}$ to be constructed local to the position, scale and time defined by $(\bm{X},r,t)$. The two postulates of refined similarity Kolmogorov (1962), known as K62, can then be formulated as follows for some randomly oriented $\bm{r}$ such that $r\ll L$ (Stolovitzky et al., 1992; Iyer et al., 2017): (i) the distribution of $\bm{V}=\Delta\bm{u}/U_{r}$ depends only upon the local Reynolds number $Re_{r}=U_{r}r/\nu$ and (ii) is independent of $Re_{r}$ when $Re_{r}\gg 1$.

Fifty-six years hence, the experimental evidence for K62 is far from conclusive and has focused exclusively on the second postulate applied to a single component of $\bm{V}$ parallel to $\bm{r}$ (Stolovitzky et al., 1992; Thoroddsen and Van Atta, 1992; Praskovsky, 1992; Thoroddsen, 1995; Qian, 1996). Early reports (Stolovitzky et al., 1992; Thoroddsen and Van Atta, 1992; Praskovsky, 1992) offered tentative support for the second postulate. However, closer inspection has revealed that the available experimental data are inconsistent with the implications of combining the second K62 postulate with three plausible models for the distribution of ${\epsilon}_{r}$ (Qian, 1996). The discrepancy lies in the use of two simplifications used to obtain ${\epsilon}_{r}$ experimentally, wherein volume averaging is replaced by one-dimensional (1D) line averaging and a 1D surrogate $\epsilon^{\prime}=15\nu(\partial u_{1}/\partial x_{1})^{2}$ is substituted for $\epsilon$. The use of the surrogate $\epsilon^{\prime}$ severely distorts the available experimental evidence, since its use weakens the dependence between $\Delta\bm{u}$ and $(r\epsilon_{r})^{1/3}$ (Wang et al., 1996) and the dependence all but disappears when other, plausible surrogates for $\epsilon$ are used (Thoroddsen, 1995; Chen et al., 1995). One or both of these simplifications have also been employed in numerical studies on the K62 postulates (Chen et al., 1993, 1995; Wang et al., 1996; Schumacher et al., 2007). Two notable exceptions are Refs. (Iyer et al., 2015) and (Iyer et al., 2017). These have provided the first evidence for K62 scaling obtained by direct numerical simulation (DNS) of the Navier-Stokes equations using 3D averages and argue that previous numerical evidence disfavouring the K62 postulates (Schumacher et al., 2007) stems from the inappropriate use of 1D averaging. The question therefore arises whether the same distribution of $\bm{V}$ found in numerical experiments can also be found in nature, which invariably lacks the statistical symmetries of such simulations that may influence both the distribution of $\bm{V}$ and its scaling (Iyer et al., 2017).

In the following, we address the deficiencies of previous experiments by directly examining the first K62 postulate without resort to surrogates. This is achieved using a recently developed technique (Lawson and Dawson, 2014) to make volumetric velocity measurements capable of directly measuring ${\epsilon}_{r}$ in a volume large enough to test the first K62 posulate across a decade of scales. We complement this data with back-to-back comparisons against direct numerical simulations of homogeneous, isotropic turbulence (Cardesa et al., 2017) to test the universality of the statistics of $\bm{V}$.

We measured the turbulence in a $1\textrm{cm}^{3}$ measurement volume near the mean-field stagnation point of a von-Kármán swirling water flow (Xu et al., 2007, 2011) using Scanning Particle Image Velocimetry (PIV) (Lawson and Dawson, 2014). This volume is small in comparison to the characteristic size of the energy containing motions $L=u^{\prime 3}/\langle\epsilon\rangle\approx 77\textrm{mm}$, where $u^{\prime 2}=\langle u_{i}^{\prime}u_{i}^{\prime}\rangle/3$ is the mean-square velocity fluctuation. The Taylor microscale Reynolds number was $R_{\lambda}\approx 200$. The working fluid, deionised water, was seeded with $6\mu\textrm{m}$ diameter PMMA microspheres with specific gravity 1.22, which are $35$ times smaller than the Kolmogorov lengthscale $\eta=(\nu^{3}/\langle\epsilon\rangle)^{1/4}\approx 210\mu\textrm{m}$ and act as passive flow tracers. The flow was illuminated with a $4.7\eta$ thick laser light sheet from a $90\textrm{W}$, pulsed, Nd:YAG laser, which was rapidly scanned across the measurement volume 250 times per second using a galvanometer mirror scanner. A pair of Phantom v640 high-speed cameras recorded the forward-scattered light at $\pm 45^{\circ}$ to the sheet at $15\textrm{kHz}$ with a resolution of $512\times 512$ pixels. Each was equipped with $200\textrm{mm}$ focal length macro lenses and $2\times$ teleconverters, providing $1:2$ optical magnification and a spatial resolution of $20\mu\textrm{m}$ per pixel. For each sample, we stored five scans with 54 consecutive images each, corresponding to a spacing between parallel laser sheets of $1.3\eta$.

The distribution of tracers was tomographically reconstructed in a discretised volume of $521\times 513\times 515$ voxels using the method described in Ref. (Lawson and Dawson, 2014). The scanning method enabled us to make reconstructions with a high seeding concentration of around 1 particle per $(1.4\eta)^{3}$. Reconstructions from sequential scans were cross-correlated with a multi-pass PIV algorithm described in Ref. (Lawson and Dawson, 2014) with an interrogation window size of $3.2\eta$ and corrections applied to account for the finite acquisition time. This yielded volumetric measurements of the velocity field in a $(42\eta)^{3}$ volume on a regular grid with spacing $0.8\eta$, from which we obtained the full velocity gradient tensor and hence the dissipation field using a least-squares finite difference stencil (Raffel, 2007). We gathered samples at $4.5$ second intervals during the continuous operation of the experimental facility for $11$ days. The water temperature was maintained at $21.2\pm 0.5^{\circ}\textrm{C}$ by a heat exchanger, seeding concentration was maintained at $24$ hour intervals and scanning PIV calibration accuracy was maintained using the method in Ref. (Knutsen et al., 2017). This yielded $2\times 10^{5}$ statistically independent volumetric snapshots of the velocity and dissipation fields.

We complement our experimental dataset with statistics obtained from publicly available DNS of forced, homogeneous isotropic turbulence lasting $66$ large eddy turnover times (Cardesa et al., 2017). The pseudospectral simulation solved the incompressible, Navier-Stokes equations on a grid of $1024^{3}$ collocation points in a triply-periodic domain with a fixed energy injection rate forcing and maximum resolvable wavenumber $k_{max}\eta=2$. Whilst the Taylor microscale Reynolds number $R_{\lambda}\approx 315$ has been surpassed by other works, the long duration of this simulation allowed us to gather well converged statistics. Velocity gradients were evaluated spectrally to obtain $\epsilon$. The local dissipation rate $\epsilon_{r}$ (1) was obtained using the spectral method in Ref. (Borue and Orszag, 1996). Following (Iyer et al., 2015), triplets of longitudinal $\Delta u(\bm{X},\bm{r},t)=\Delta\bm{u}\cdot\hat{\bm{r}}$ and transverse $\Delta v(\bm{X},\bm{r},t)=\Delta\bm{u}\cdot(\bm{e}_{j}\times\hat{\bm{r}})$ velocity increments were evaluated for $\hat{\bm{r}}$ oriented in each of the three principal grid directions $\bm{e}_{i}$ ($i\neq j$) over separations $r/\eta$ of $3.0,5.9,8.9,11.8,17.8,23.7,32.6$, corresponding to logarithmically spaced, even multiples of the grid spacing. Statistics were evaluated for each grid point in $66$ snapshots of the flow field spaced evenly in time over the simulated time interval.

In contrast to the numerical simulation data, the data from our von-Kármán mixing tank exhibit a statistical axisymmetry aligned with the axis of the counter-rotating disks (Voth et al., 1998, 2002; Lawson and Dawson, 2015). We therefore adopt a careful definition of our statistical ensemble of $\Delta u(\bm{X},\bm{r},t)$, $\Delta v(\bm{X},\bm{r},t)$ and ${\epsilon}_{r}(\bm{X},r,t)$ in order to recover the isotropic scaling behaviour. For a single point $\bm{X}$ near the mean-flow stagnation point, we evaluate the longitudinal and transverse velocity increments over $2940$ orientations of the separation vector $\bm{r}$ uniformly spaced over the surface of a sphere of diameter $r$. Numerically, this is achieved using a cubic spline interpolation of the velocity and dissipation field at scales $r/\eta$ chosen from the geometric series $1.5,2.1,...36.2$. Statistics are then gathered over each of the $2\times 10^{5}$ realisations of the flow. This angle-averaging of statistics is directly related to the SO(3) decomposition (Arad et al., 1998; Biferale and Procaccia, 2005), which enables the recovery of isotropic scaling properties in flows with statistical anisotropies (Arad et al., 1999; Kurien et al., 2000).

To test the first K62 postulate, we consider the conditional expectations of the form

[TABLE]

Under the first K62 postulate, these conditional averages should only depend upon $Re_{r}$.

Figure LABEL:fig:DLr-cond-Rer shows the conditional average (2) of the magnitude of the longitudinal velocity increment given the local Reynolds number based on $U_{r}$. At comparable scales $r/\eta$, the experimental and numerical data are in close, quantitative agreement. For each curve with fixed $r$ and $\nu$, we are effectively examining the conditional expectation of $|\Delta u|$ for different local characteristic velocity scales $U_{r}$. At small $Re_{r}$ the data are in close agreement with the scaling $|\Delta u|r\sim Re_{r}^{3/2}$, which is expected from a Taylor series expansion at small $r$ (Wang et al., 1996). At larger $Re_{r}$, the scaling approaches $|\Delta u|r\sim Re_{r}$, which is expected from the second K62 postulate. If the first postulate holds exactly, given that $r\ll L$, we should expect that (2) only depends on $Re_{r}$. Instead, we notice that a systematic dependence upon the scale $r$ is retained, which becomes less significant as the local Reynolds number is increased.

In contrast, Figure LABEL:fig:DNr-cond-Rer shows the equivalent conditional average (3) for the transverse velocity increments. Again, there is excellent agreement between numerics and experiment. Good collapse across scale $r$ is observed for $Re_{r}\gtrsim 10$. At smaller $Re_{r}$, the collapse across scale is less compelling. This may be anticipated from a consideration of the limiting behaviour of $V_{T}$ at small $r$. Based on a Taylor series expansion of $\Delta v$ with orientation averaging, we obtain \langle V_{T}^{2}|Re_{r}\rangle=Re_{r}/20+Re_{r}\langle\Omega/\epsilon\big{|}Re_{r}\rangle/12, where $\Omega$ is the enstrophy $\Omega=\nu(\nabla\times\bm{u})^{2}$. It follows that in the limit of $r\rightarrow 0$, \langle\Omega\big{|}\epsilon\rangle must scale linearly with $\epsilon$ for \langle V_{T}^{2}\big{|}Re_{r}\rangle to depend only upon $Re_{r}$. Such a linear scaling has been shown to hold in relatively active dissipative regions $\epsilon>\langle\epsilon\rangle$ of homogeneous isotropic turbulence, but breaks down for $\epsilon\ll\langle\epsilon\rangle$ (Donzis et al., 2008). The discrepancy may be resolved as the Taylor microscale Reynolds number is increased (Yeung et al., 2015).

As a detailed, direct test of first postulate, Figure 2 shows the conditional distribution of $V_{L}$ and $V_{T}$ given the local Reynolds number over different spatial scales $r$. For both longitudinal and transverse increments, we observe good, quantitative agreement between experimental and numerical data at comparable scales $r/\eta$ and matched local Reynolds numbers. We first consider the longitudinal velocity increment. The left panels of Figure 2 demonstrate that the distribution of $V_{L}$ largely collapses across scale when conditioned upon the local Reynolds number. The collapse improves as the local Reynolds number is made larger. The scale dependence of the conditional distribution of $V_{L}$ appears to be stronger in our data than the numerical simulation results of Iyer et al. (2015). This may be due to the smaller scale separation $r/L$ achieved in the present experimental study. We offer an additional remark that, when $U_{r}$ is instead based on local averages of the pseudodissipation $\phi=\nu A_{ij}A_{ij}$, an improved collapse is observed for the longitudinal velocity increment.

The right panels of Figure 2 show the equivalent conditional distribution for the transverse velocity increment. For fixed $Re_{r}$, the transverse increments show an improved collapse across scale in comparison to their longitudinal counterparts. This confirms the approximate validity of the first refined similarity hypothesis for transverse velocity increments.

The application of scanning PIV has allowed us to directly examine the first K62 postulate in a laboratory flow using three-dimensional, local averages of the dissipation, thereby resolving the surrogacy issue which has confounded previous experimental investigations. We have complemented our experimental analysis with back-to-back comparisons against high-resolution DNS of homogeneous isotropic turbulence. We observe that the distributions of $V_{L}$ and $V_{T}$ and their average magnitudes are in close agreement between both flows when the local Reynolds number and scale are matched. The first postulate is shown to approximately hold for both longitudinal and transverse increments, with improved agreement found for larger local Reynolds numbers. Our study provides the first unambiguous experimental evidence to demonstrate that a detailed, universal description of high Reynolds number turbulence may at last be within grasp.