Initial-separation window, Richardson constant, and objective inertial-range criteria for Richardson $t^3$ two-particle dispersion at finite Reynolds number

Whether, at what initial separations, and with what universal constant the Richardson–Obukhov law $\langle|\mathbf{r}(t)-\mathbf{r}(0)|^2\rangle = g\langle\epsilon\rangle t^3$ can be observed at finite Reynolds number remains contested. The highest-Reynolds-number Lagrangian data ($R_\lambda=1000$) have been read three incompatible ways: a single critical initial separation $r_0/\eta\approx 3$, a window $2<r_0/\eta<8$ that widens with $R_\lambda$ with $g\to 0.55$, or a crossover artifact between Batchelor and Taylor dispersion regimes with no genuine $t^3$ range at inertial-range $r_0$. The open problem is to determine how the initial-separation window exhibiting Richardson scaling depends on Reynolds number, whether the cubed-local-slope tendency to plateau at $g \approx 0.55$ is confirmed as $R_\lambda$ increases, and whether inertial-range exponents can be reliably extracted by fixed-window power-law fits or require plateau-based local-slope diagnostics.

Background

Two-particle (pair) dispersion in turbulence describes how the separation vector $\mathbf{r}(t)$ between two fluid particles grows in time, and it underpins the theory of turbulent mixing. Richardson's 1926 atmospheric observations, together with the dimensional-analysis derivation given by Obukhov in 1941 within Kolmogorov's inertial-range framework (a history surveyed in the review of Salazar and Collins), predict that, for initial separations inside the inertial range ($\eta \ll r_0 \ll L$, where $\eta$ is the Kolmogorov dissipation length and $L$ the large scale of the flow) and at intermediate times $t_0 \ll t \ll T_L$, the mean-square separation grows cubically: $\langle|\mathbf{r}(t)-\mathbf{r}(0)|^2\rangle = g\,\langle\epsilon\rangle\,t^3$, where $\langle\epsilon\rangle$ is the mean energy dissipation rate per unit mass, $g$ is the dimensionless Richardson constant, $t_0=(r_0^2/\langle\epsilon\rangle)^{1/3}$ is the Batchelor timescale associated with memory of the initial separation $r_0$, and $T_L$ is the Lagrangian integral timescale marking the transition to diffusive (Taylor) growth. At early times $t \ll t_0$ the pair still remembers $r_0$ (the Batchelor regime), and at late times $t \gg T_L$ the mean-square separation grows linearly in time (the Taylor, diffusive regime); Richardson's law can hold only in the intermediate window. The Taylor-scale Reynolds number $R_\lambda$, formed with the root-mean-square velocity fluctuation and the Taylor microscale (the length defined by the curvature at the origin of the spatial velocity autocorrelation), is the standard measure of the separation between the large and dissipative scales of the flow. By analogy with the Kolmogorov constant of the energy spectrum, $g$ is expected to be a universal constant, but reported values scatter over roughly $0.5$–$0.7$. Observing the law is hard because the temporal inertial range $T_L/t_\eta$ ($t_\eta$ the Kolmogorov timescale) is much narrower than the spatial one, and because $T_L/t_0 = (T_L/t_\eta)(r_0/\eta)^{-2/3}$ shrinks further for $r_0/\eta > 1$. The two highest-Reynolds-number direct numerical simulation (DNS) datasets with Lagrangian particle tracking reach $R_\lambda = 650$ and $1000$ (Buaria, Sawford and Yeung, Phys. Fluids 27, 105101 (2015); and Buaria, Yeung and Sawford, J. Fluid Mech. 799, 352 (2016), the latter adding molecular (Brownian) diffusion at Schmidt number $Sc = 1$, the ratio of viscosity to molecular diffusivity, which accelerates the loss of initial-separation memory). From laboratory experiments at lower $R_\lambda$ together with published data, Tan and Ni (Phys. Rev. Lett. 128, 114502 (2022)) fitted a power law $\langle|\mathbf{r}(t)-\mathbf{r}(0)|^2\rangle \sim t^{k_D}$ over the window $2t_0 \le t \le T_L/2$ and concluded that cubic scaling ($k_D = 3$) occurs at finite $R_\lambda$ only for one critical initial separation $r_0 \approx 3\eta$, and for any $r_0$ only at infinite $R_\lambda$. Buaria's Comment (Phys. Rev. Lett. 130, 029401 (2023)) showed, using the logarithmic local slope (the running exponent $d\log\langle|\Delta\mathbf{r}|^2\rangle/d\log t$) and the cubed-local-slope (CLS) diagnostic $\frac{1}{\langle\epsilon\rangle}\left(\frac{d}{dt}[\langle|\mathbf{r}(t)|^2\rangle]^{1/3}\right)^3$, which plateaus at $g$ when Richardson scaling holds, that the ad-hoc fitting window cannot identify a unique exponent; plateau-based extraction instead gives $k_D = 3$ for $2 < r_0/\eta < 8$ (effective $r_0/\eta \approx 6$) at $R_\lambda = 1000$, a clear CLS tendency toward $g \approx 0.55$ (most robust for $Sc=1$ diffusive particles), and trends suggesting that the window of $r_0/\eta$ exhibiting cubic scaling widens with $R_\lambda$ — though at $R_\lambda = 1000$ even $r_0/\eta = 8$ affords only $T_L/t_0 \approx 20$, barely one decade. Meanwhile Elsinga, Ishihara and Hunt (J. Fluid Mech. 932, A17 (2022)) surveyed the literature and found that $t^3$ scaling has never been observed for initial separations genuinely inside the inertial range; they argue that the observed intermediate regime is a transition region connecting the Batchelor and Taylor regimes, that apparent $t^3$ at $r_0/\eta \approx 4$ is a coincidence of this connection, and they propose objective requirements for identifying Richardson scaling: $r_0$ larger than the size of dissipative structures (about $60\eta$, conservatively $120\eta$), persistence over at least a decade of time, and caution with virtual-origin fits — fits of $\langle|\mathbf{r}(t)-\mathbf{r}(0)|^2\rangle \propto (t - t_{\mathrm{v}})^3$ with a free time offset $t_{\mathrm{v}}$ — which can mimic $t^3$ over up to half a decade. In their Reply (Phys. Rev. Lett. 130, 029402 (2023)) Tan and Ni defended the single-critical-separation conclusion, noting that Buaria's own tabulated exponents cross $k_D=3$ at a single bin of $r_0/\eta$ and maintaining that their conclusion does not depend on the method chosen to extract the scaling exponent; both sides agreed that Reynolds numbers beyond the currently available ones are needed to settle the question, the Reply pointing to large-scale experimental campaigns in geophysical flows as one possible way to put the study to the test. The Comment and Reply thus jointly pose, but leave open, the empirical and methodological problem formulated below.

Problem Statement

Determine the finite-Reynolds-number observability conditions and content of Richardson's law for two-particle dispersion in incompressible turbulence. Concretely, the problem — as jointly posed by Buaria's Comment (Phys. Rev. Lett. 130, 029401 (2023)) and left open by the Comment/Reply exchange with Tan and Ni — comprises three scientifically linked parts: (1) quantify how the range of initial separations $r_0/\eta$ that exhibits robust Richardson $t^3$ scaling (mean-square separation $\langle|\mathbf{r}(t)-\mathbf{r}(0)|^2\rangle = g\langle\epsilon\rangle t^3$) at finite Taylor-scale Reynolds number $R_\lambda$ depends on $R_\lambda$ — deciding among the competing readings of the existing data (a single critical $r_0/\eta\approx 3$; a window, currently $2<r_0/\eta<8$ at $R_\lambda=1000$, that widens with $R_\lambda$; or a Batchelor-to-Taylor crossover artifact with no genuine $t^3$ range for inertial-range $r_0$), and in particular determine what $R_\lambda$ is required to obtain multiple decades of temporal scale separation $T_L/t_0$ for representative $r_0/\eta>1$; (2) determine whether the cubed-local-slope (CLS) tendency to plateau at $g \approx 0.55$ for various $r_0/\eta$ bins observed at the highest available $R_\lambda$ (most robust for $Sc=1$ diffusive particles) is confirmed as $R_\lambda$ increases — i.e. whether the CLS plateaus at the same $g$ for various $r_0/\eta$ as Richardson scaling requires — given that reported values of $g$ scatter over roughly $0.5$–$0.7$; and (3) resolve the methodological disagreement posed by the exchange: whether inertial-range exponents can be reliably extracted by fixed-window power-law fits over $2t_0 \le t \le T_L/2$ with error bars (Tan and Ni's position, who maintain that their conclusion does not depend on the method chosen to extract the scaling exponent), or whether such a window cannot identify a unique exponent and plateau-based local-slope/CLS extraction is required instead (Buaria's position). Both sides agree that significantly higher $R_\lambda$ than currently available is required to obtain a sufficient scale separation ($T_L/t_0 \approx 20$ at $R_\lambda = 1000$ for $r_0/\eta = 8$, barely one decade); the Reply cites large-scale experimental campaigns in geophysical flows as one possible way to obtain the needed measurements. All three parts are mandatory and must each be answered; together they share one objective: establishing robustly which initial separations exhibit Richardson scaling at finite Reynolds number, by which diagnostic that scaling is reliably established, and what value the Richardson constant $g$ carries.

References

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