Implications of inertial subrange scaling for stably stratified mixing

Homogeneous sheared and stratified turbulence, HSST, is assumed to satisfy the Navier-Stokes equations subject to the non-hydrostatic Boussinesq approximation. The dimensional equations for the fluctuations relative to the planar means are

	$\displaystyle\frac{\partial\mathbf{u}}{\partial t}+(\mathbf{u}\cdot\nabla)\mathbf{u}$	$\displaystyle=-\frac{1}{\rho_{0}}\nabla p-z\frac{\partial\mathbf{u}}{\partial x}\frac{d\bar{u}_{x}}{dz}-u_{z}\frac{d\bar{u}_{x}}{dz}\hat{\mathbf{x}}-\frac{g_{z}}{\rho_{0}}\rho\hat{\mathbf{z}}+\nu\nabla\cdot\nabla\mathbf{u},$		(4a)
	$\displaystyle\frac{\partial\rho}{\partial t}+(\mathbf{u}\cdot\nabla)\rho$	$\displaystyle=-z\frac{\partial\mathbf{\rho}}{\partial x}\frac{d\bar{u}_{x}}{dz}-u_{z}\frac{d\bar{\rho}}{dz}+D\nabla\cdot\nabla\rho,$		(4b)
	$\displaystyle\nabla\cdot\mathbf{u}$	$\displaystyle=0,$		(4c)

where $\mathbf{u}=(u_{x},u_{y},u_{z})$ is the velocity vector in the coordinate system $(x,y,z)$ and functional dependencies have been dropped for convenience, and $\rho$ is the fluctuating density field so that the total density is

$$\rho_{t}=\rho_{0}+\overline{\rho}(z)+\rho(x,y,z,t),$$

(5)

with $\overline{\rho}(z)=zd\overline{\rho}/dz$ being the ambient density that is constant in time and has a uniform gradient antiparallel to gravitational acceleration $g_{z}$. Similarly, $\bar{u}_{z}=zd\overline{u}_{x}/dz=zS$ is the mean velocity and $S\equiv d\bar{u}_{x}/dz$ is the mean shear. The pressure decomposition is analogous to that of density and velocity so that $p$ is the fluctuating mechanical pressure relative to the temporally-constant planar mean. Internal energy and scalar transport affecting density have been combined into a single evolution equation for $\rho$ with $D$ the molecular diffusivity, $\hat{\mathbf{x}}$ and $\hat{\mathbf{z}}$ are the unit vectors in the $x$- and $z$-directions respectively, and $\nu$ is the kinematic viscosity.

Local isotropy is a state wherein statistical symmetries of multi-point statistics, such as homogeneity, isotropy and stationarity, are present in a spatio-temporal localised region (Monin & Yaglom, 1975). In the presence of anisotropic integral scales, the conditions of local isotropy are defined, in the spatial sense, by a quasi-equilibrium range of scales that are sufficiently smaller than integral scales (Kolmogorov, 1941).

This is a prerequisite of classical inertial subrange scaling arguments (Kolmogorov, 1941; Oboukhov, 1941a; Corrsin, 1951). These state that scales within the quasi-equilibrium regime exist wherein turbulence is independent of the effects of viscosity. In shear flows, conditions for the application of local isotropy are thought to be valid at scales smaller than $L_{C}$ (Corrsin, 1958; Uberoi, 1957). Kolmogorov scaling (Kolmogorov, 1941, 1962) in the presence of anisotropic outer-scales has been revealed to coincide at scales consistent with local isotropy (Champagne et al., 1970; Saddoughi & Veeravalli, 1994). The equivalent local isotropy justification of the Kolmgorov subrange may be applied to Obukhov-Corrsin passive scalar scaling (Oboukhov, 1949; Corrsin, 1951) such that it is fit to describe an active scalar below scales affected by stratification (Monin & Yaglom, 1975, pg 391). Therefore, the spectral density of potential energy is expected to be determined by the energy dissipation rates and a wavenumber, i.e.

$$\hat{E}_{p}^{OC}(|\mathbf{k}|)=\beta\chi\epsilon^{-1/3}|\mathbf{k}|^{-5/3}f_{L}^{OC}(|\mathbf{k}|L_{C})f_{v}^{OC}(|\mathbf{k}|L_{B}),$$

(17)

where $\mathbf{k}$ is the wavenumber vector, $\beta$ is the Obukhov-Corrsin constant, the functions denoted by $f^{OC}_{v}$, $f^{OC}_{L}$ are the viscous and outer scale correction functions that are necessary without applying assumptions of local isotropy. Furthermore, according to Oboukhov (1941a), the kinetic energy spectra

$$\hat{E}_{k}^{K}(|\mathbf{k}|)=C\epsilon^{2/3}|\mathbf{k}|^{-5/3}f_{L}^{K}(|\mathbf{k}|L_{C})f_{v}^{K}(|\mathbf{k}|L_{K}),$$

(18)

where $C$ is the Kolmogorov constant, and $f^{K}_{L}$ and $f^{K}_{v}$ are the correction functions. Following the analysis of Beguier et al. (1978), by integration of (17) and (18) when $L_{C}\gg|\mathbf{k}|^{-1}\gg L_{K},L_{B}$, it is possible to obtain

$$\Pi\equiv\frac{E_{p}\epsilon}{E_{k}\chi}=\frac{\beta}{C}.$$

(19)

Therefore, the explicit relation for modeling the irreversible flux from (19) is

$$j_{b}\approx\chi\approx\frac{E_{p}\epsilon}{E_{k}\Pi},$$

(20)

with the limiting behaviour, at high Reynolds number, that $\Pi\rightarrow\beta/C$. A critical simplifying assumption in the development of (20) is that ${Fr}\geq O(1)$, though the imposition of more complex scaling relations which are thought to develop when ${Fr}\ll O(1)$ may alternatively be imposed. Indeed, more generally,

$$\Pi=\left(\frac{L_{OC}}{L_{LE}}\right)^{2/3}=\frac{N_{\ast}}{{S_{\ast}}\sqrt{{Ri}}}=N_{\ast}{Fr}\quad.$$

(21)

We stress that the assumed model spectra (17) and (18) are approximations which are difficult to observe precisely in carefully controlled experiments or simulations. Whereas these two-point scaling models are observed in flows with anistropic large scales, they are not claimed generally applicable for flows in arbitrary parameter regimes. However the sensitivity of $\Pi$ with respect to measured deviations from the model spectra is unclear a priori, a point we discuss in §4.

The relation (20) has profound implications for the calibration of a number of turbulent stratified mixing models. Generally, in gradient diffusion models,

$$\kappa_{b}\approx\frac{E_{p}}{\Pi N}{Fr},$$

(22)

while gradient diffusion subject to the model proposed by Osborn (1980) leads to

$$\Gamma_{b}\approx\Gamma\equiv\frac{\chi}{\epsilon}=\frac{E_{p}}{E_{k}\Pi}\text{ ,}$$

(23)

Combining the turbulent viscosity model of Crawford (1982) with the turbulent diffusivity model of Osborn (1980), the turbulent Prandtl number may then be expressed as

$${Pr_{t}}=\frac{1+\Gamma_{b}}{\Gamma_{b}}{Ri}=\frac{E_{k}\Pi+E_{p}}{E_{p}}{Ri}\text{ .}$$

(24)

Alternatively, the relevance of the parameter $\Pi$ to the turbulent Prandtl number model of Venayagamoorthy & Stretch (2010) implies that their key mixing parameter $\gamma$ may be expressed as

$$\gamma=\frac{\Pi}{2}\quad,$$

(25)

where $\gamma$ too has been empirically observed to assume relatively insensitive values at moderate Reynolds numbers in HSST (Venayagamoorthy & Stretch, 2006). Finally, if the objective is to construct a mixing length scale for applications in Prandtl mixing length models, such as the framework of Odier et al. (2009), where $\kappa_{b}=l_{m}^{2}S$, (20) may be equivalently expressed as

$$l_{m}=\frac{E_{p}}{\Pi N^{2}}{S_{\ast}}\text{ .}$$

(26)

The homogeneous stratified shear system defined by (4a) and (4b) is solved in a triply periodic domain with a Fourier pseudo-spectral method. Timestepping is performed with a fractional step method where non-linear terms are advanced with a third-order Adams-Bashforth scheme. Aliasing is treated by a sharp-spectral filter at $15/16k_{max}$, which proved to be sufficient in a-posteriori analysis. Temporal integration of the inhomogeneous shear term is handled by the integrating factor approach of Sekimoto et al. (2016). The rest of the solver is derived from the method used in Hebert & de Bruyn Kops (2006), with further details discussed therein.

The suite of simulations are designed to resolve sufficiently both the smallest and largest spatiotemporal turbulent flow scales. The streamwise extent, $L_{x}$, is is chosen to be 40 times larger than the large-eddy length scale. Because of the anisotropy of integral length scales that was observed in a limited parameter space study, the cross-stream domain length $L_{y}$ and the vertical domain length $L_{z}$ are chosen such that $L_{x}/L_{y}=2$ and $L_{x}/L_{z}=4$. A visualisation of the computational domain is featured in figure 1.

The smallest spatial scales are resolved by enforcing $k_{max}L_{K}\approx 2$, where $k_{\text{max}}$ indicates the maximum wavenumber supported by the domain discretisation when fully-resolved.

Finally, we have adopted a mass-spring-damper-like system (as similarly adopted in Overholt & Pope (1998); Rao & de Bruyn Kops (2011)), to tune the Richardson number to its stationary value by setting a constant kinetic energy target $E_{t}$. We note that this is similar to the constant power criteria as used in stationary HSST by Chung & Matheou (2012) but that here we include damping. The system is controlled by time-dependent variation of $Ri$ being required to satisfy the equation

$$c_{0}S{Ri}^{\prime}(t)+2\alpha\omega\tilde{E}_{k}^{\prime}(t)+\omega^{2}(\tilde{E}_{k}(t)-1)=0,$$

(27)

where the prime notation denotes a temporal derivative, $\tilde{E}_{k}(t)\equiv E_{k}/E_{t}$ is the normalised turbulent kinetic energy, $\omega$ is the characteristic frequency of oscillation, $\alpha$ is a dimensionless damping factor, and $c_{0}$ is a dimensionless parameter. This control system has been derived by assuming that the kinetic energy follows a second order linear system (e.g. Rao & de Bruyn Kops, 2011), and then by applying a first-order approximation to the temporal evolution of kinetic energy (c.f. Jacobitz et al., 1997)

$$\tilde{E}_{k}^{\prime}(t)\approx c_{0}S({Ri}(t)-{Ri}_{c})\hbox{\ such that \ }\tilde{E}_{k}^{\prime\prime}(t)\approx c_{0}S{Ri}^{\prime}(t).$$

(28)

The choice for the parameter $c_{0}\approx-1$ is supported by Jacobitz et al. (1997), the characteristic frequency $\omega$ is determined by the large-eddy time scale which itself is proportional to $1/S$ (Shih et al., 2000), and finally a damping coefficient $\alpha=1.5$ was found to work well.

The stationarity constraint and the emergent empirical observation that $S_{\ast}\approx 5$ lead to the natural consideration of sets of simulations which can test the effects of variation of the Reynolds number, essentially independently of the other parameters. The simulations presented here are largely equivalent to those reported in Portwood et al. (2019), though some cases have been run for longer times to ensure convergence of statistics. The various parameters of the simulations are summarised in Table 1, and some flow visualisations of the density field are shown in figure 1. Two key aspects are apparent, as noted in the caption. First, in the horizontal plane visualisations, as the Reynolds number increases, the magnitude of the density fluctuations tends to decrease, while, unsurprisingly the dynamic range of scales increases, with enhanced smaller scale structures. Second, in the vertical plane visualisation, inclined large-scale structures are apparent, consistently with the results of Chung & Matheou (2012) and Jacobitz & Moreau (2016).

Time series are shown in figure 2 to illustrate convergence. Richardson numbers tend to fluctuate within about 10% of their means, at a confidence interval at least as precise as critical Richardson numbers reported by Jacobitz et al. (1997) and Shih et al. (2000). The fluctuations in the Richardson number occur at large integral timescales such that effects of $Ri^{\prime}(t)$ are small. Energetic fluctuations are more substantial than higher dimensional forcing schemes (i.e. Overholt & Pope, 1998; Rao & de Bruyn Kops, 2011) that dynamically control the flow with more granularity by accessing a broader range of turbulent length scales compared to a single mean-scale parameter as is done here. Nonetheless, energy remains approximately stationary over the large time scales of the simulations.

Recalling that the critical Richardson number is an emergent quantity from the control system defined by (27), and is not determined a priori, an increase of the critical Richardson number is not observed in these simulations as suggested by Holt et al. (1992); Shih et al. (2000), although the emergent critical value of the Richardson numbers are broadly consistent with those reported by (Shih et al., 2000). Specifically, we observe the critical gradient Richardson number which induces statistical stationarity in these flows to be approximately 0.16. Similarly, we observe ${Fr}\approx 0.5$ in all simulations, which implies ${S_{\ast}}\approx 5$, as observed in other studies (Shih et al., 2000; Jacobitz et al., 1997).

Curiously, the lowest Reynolds number to maintain stationarity robustly according to our stringent criteria corresponds approximately to case R1. Case R1 corresponds to ${Re_{b}}\approx 36$, which is consistent with critical values observed and estimated for sustained three-dimensional turbulence previously (Gibson, 1980; Shih et al., 2005; Portwood et al., 2016).

For further comparison with the flows considered by Shih et al. (2000) and Shih et al. (2005), we plot their relevant cases against the solutions presented for this research in a $Fr-Re_{b}$ parameter space as shown in figure 3. We remark that the configurations reported by Shih and co-authors in those publications feature transitional and, crucially, variable $Fr$ when $Re_{b}$ is $O(100)$. As is apparent in the figure, $Fr$ and $Re_{b}$ appear to be coupled, with approximate scaling relationship $Fr\propto Re_{b}^{1/2}$, as shown with a dashed line. Therefore, the parameterisation of mixing by $Re_{b}$ cannot be made independent of $Fr$ with the data presented in Shih et al. (2000, 2005). We note that this explanation does not rely on any argument that the simulations are inadequately resolved, either at small or large scales. Our perspective is qualitatively different from the perspective presented in Kunze et al. (2012); Gregg et al. (2018), which suggests the $Re_{b}$-sensitive regime with $Re_{b}>100$ is due to inadequate small-scale spatial resolution in the simulations. However, it is entirely plausible that the solutions analysed in Shih et. al. suffer from inadequate large scale resolution in the domain as suggested by other reports (see Kunze, 2011; Kunze et al., 2012; Gregg et al., 2018).

Therefore, disentangling the behaviour of their simulations as $Re_{b}$ varies from the effects of variation in $Fr$ is challenging, particularly when $Re_{b}$ (or $Re_{s}$) is large. It is important to remember that it was precisely for $Re_{b}\gtrsim 100$ that Shih et al. (2005) argued that the turbulent flux coefficient $\Gamma\propto Re_{b}^{-1/2}$, whereas it is entirely plausible that variable $Fr$ is playing a significant role. This figure highlights the novelty of our simulations in this particular flow configuration, which allow the isolated study of the effects of variations in Reynolds number from the influence of other non-dimensional turbulent parameters, and in particular at constant $Fr$.

Just as non-dimensional parameters are emergent quantities in our simulations, so too are the partitionings of potential energy and the various components of kinetic energy. Such partitionings are significant, not least because the ratio of potential energy to kinetic energy,

$$R_{PK}\equiv E_{p}/E_{k}\;\text{ ,}$$

(29)

is a critical component to mixing models wherein Reynolds number, or ${Re_{b}}$, dependence is often omitted and the mixing is assumed to be a function of ${Ri}$ (Osborn & Cox, 1972; Schumann & Gerz, 1995; Pouquet et al., 2018). Furthermore, in ‘strongly’ stratified turbulent flow, Billant & Chomaz (2001) suggests that there should be approximate equipartition between potential and kinetic energy, i.e. $R_{PK}\approx 1$, an assumption also used by Lindborg (2006). It is important to remember that these models do not account for Reynolds number effects. Therefore, parameterisation of such energetic partitioning is an important a priori validation necessary for model evaluation.

Recalling that the kinetic energy is the same for all cases, by construction from our definition of stationarity, we observe a significant decrease in potential energy with increasing dynamic range as shown in figure 4b. In comparison, the results of Brethouwer et al. (2007), which span up to ${Re_{b}}\sim O(10)$ at much lower Froude number in homogeneous stratified turbulence, report $R_{PK}\approx 0.15$ at ${Re_{b}}={Ri}{Re_{s}}\approx 16$, noting an asymptotic trend to this value from $R_{PK}\approx 0.05$ at ${Re_{b}}\approx 0.1$. Though we observe similar values at ${Re_{s}}\approx 5$ (corresponding to ${Re_{b}}\approx 30$), higher values of ${Re_{b}}$ reveal a subsequent decline in $R_{PK}$, as suggested by Remmler & Hickel (2012). For Reynolds numbers larger than ${Re_{s}}\approx 40$ (${Re_{b}}\approx 300$), $R_{PK}$ assumes an apparently asymptotic value of $0.08$ in these simulations. This relative decrease of potential energy as a function of ${Re_{s}}$ can be observed in the domain slice visualisations of $\rho$ featured in figure 1, as there is a clear reduction in the variance of the density fluctuations. In terms of the important question concerning the modeling of mixing, and in particular parameterising the turbulent flux coefficient $\Gamma$, (23) shows that if $R_{PK}$ approaches an asymptotic value for large $Re_{s}$, $\Gamma$ can exhibit parameter dependence only if the key parameter $\Pi$ (defined in (19)) does.

Furthermore, a distinguishing characteristic of sheared stratified turbulence, relative to other stratified flows, is the absence of axisymmetry normal to gravity with the result that anisotropy must be parameterised in each dimension. Inertially-relevant anisotropy can be characterised by the variance of individual velocity vector components as they contribute to the mean turbulent kinetic energy. These velocity variances resulting from anisotropic dynamics are shown in figure 4a.

We observe turbulent kinetic energy is dominated by streamwise velocity fluctuations with the smallest contributions coming from vertical velocity fluctuations. Notably distinct from axisymmetric flows, the horizontal components of velocity variance represent approximately 80 percent of the total contributions to kinetic energy. The cross-stream velocity variance accounts for only 30 percent of the total energy. The vertical variance, subject to exchanges to potential energy, is typically suggested to vary as a function of Froude number (Brethouwer et al., 2007). Here, the vertical variance accounts for approximately 20 percent of the total energy and we observe that it remains largely constant as a function of ${Re_{b}}$.

Perhaps surprisingly, anisotropy of velocity fluctuations increases with increasing dynamic range. That is, the streamwise velocity component becomes increasingly dominant with increasing Reynolds number, apparently at the expense of the cross-stream velocity variance. One possible interpretation is that reducing viscous effects at inertial scales as the dynamic range increases allows the flow to evolve towards an inertial equilibrium state.

It might be assumed that the significant energetic transitions at sufficiently high Reynolds number would be explained by accompanying clear transitions in dynamics. Here we analyse the stationary behaviour of the dynamics relevant to the kinetic and potential energy balances. The ensemble-averaged energy equations, as derived from (4), obey

	$\displaystyle\left\langle\frac{\partial{E_{p}}}{\partial t}\right\rangle$	$\displaystyle=B-\chi\;\text{ and }$		(30)
	$\displaystyle\left\langle\frac{\partial{E_{k}}}{\partial t}\right\rangle$	$\displaystyle=P-B-\epsilon\;\text{ ,}$		(31)

where the turbulent production from mean shear $P=-\langle u_{x}u_{z}\,d\bar{u}_{x}/{dz}\rangle$, the ‘buoyancy’ flux $B=\langle({g}/{\rho_{0}}){u_{z}}\rho\rangle$ have anisotropic effects on the evolution of the streamwise and vertical components of the kinetic energy $\langle u_{x}^{2}\rangle$ and $\langle u_{z}^{2}\rangle$ respectively.

Note that the left hand sides of (31), (30) reduce to approximately zero due to the stationarity constraint. Therefore, the dynamics should, at least in principle, be describable by the turbulent flux coefficient $\Gamma$ defined in terms of $\chi$ in (23), and a flux Richardson number, defined as

$${Rf}\equiv B/P\simeq\frac{\chi}{\chi+\epsilon}=\frac{\Gamma}{1+\Gamma},$$

(32)

since, if the flow actually enters an emergent stationary state the flux $B\simeq\chi$, and remembering the underlying original definition (3) $\Gamma_{b}\simeq\Gamma$. Furthermore, in this circumstance, the turbulent viscosity $\nu_{b}$ can be naturally defined as

$$\nu_{b}\equiv\frac{P}{S^{2}}\rightarrow{Pr_{t}}\equiv\frac{\nu_{b}}{\kappa_{b}}\simeq\frac{{Ri}}{{Rf}}\simeq\left(\frac{1+\Gamma}{\Gamma}\right){Ri},$$

(33)

and so we can obtain an expression consistent with (24). Henceforth, we continue to describe the evolution of energetics by $\Gamma$ defined as in (23) in recognition of the longstanding application of different definitions of the turbulent flux coefficient to estimating turbulent diffusivity and turbulent viscosity (Osborn & Cox, 1972; Crawford, 1982). Gregg et al. (2018) discuss in detail the subtleties and potential for confusion concerning different definitions for turbulent flux coefficients, and so it must always be remembered that here we concentrate on using the definition in (23), i.e. $\Gamma\equiv\chi/\epsilon$.

The ratio of terms in the energetic balance is shown in figure 5. We observe a slight decline of $\Gamma$ from $0.2$ to approximately $0.175$, as noted in Portwood et al. (2019), until ${Re_{s}}\approx 40$. However, $\Gamma$ remains approximately constant as $Re_{s}$ increases further, and in particular we do not observe the proposed ${Re_{b}}^{-1/2}$ scaling reported in a similar flow by Shih et al. (2005) for their energetic regime of ${Re_{b}}>100$. Remembering the data presented in figure 3, this is perhaps unsurprising due to the apparent correlated variation of $Fr$ and $Re_{b}$ in those simulations.

Due to the condition of stationarity being imposed on the kinetic energy in our simulations, accompanying induced stationarity of the potential energy is also guaranteed, though still plotted for reference in figure 5 demonstrating that $\chi\approx B$. It is also apparent that ${Rf}\equiv B/P$ is less than $\Gamma\equiv\chi/\epsilon$, as expected.

Since we have observed ${Fr}$ and ${Ri}$ to be essentially independent of the Reynolds number, the relatively decreasing potential energy as ${Re_{s}}$ increases for $Re_{s}\lessapprox 50$ implies that scales characteristic of the scalar should also exhibit a similar asymptotic trend. The scalar outer-scale associated with Obukhov-Corrsin similarity, $L_{OC}$ is defined in (14). In figure 6a, we plot the ratio of this outer-scale to the Ozmidov length scale $L_{O}$, as defined in (7).

As $Re_{s}$ increases, this ratio decreases for $Re_{s}\lessapprox 40.$ For larger $Re_{s}$, there is some evidence that the ratio remains constant near unity, suggesting of course that $L_{OC}$ anomalously scales with $L_{O}$. This implies that the scaling $\chi\approx E_{p}N$ is valid in this high Reynolds number regime and that outer-scales of the scalar adjust to the regime associated by local isotropy, which is indeed a condition of a Obkhov-Corrsin subrange in anisotropic flows.

Length scales associated with the mixing length models suggested by Odier et al. (2009) can also be shown to remain constant in our large $Re_{s}$ regime, and indeed even for smaller $Re_{s}$. The scalar mixing length, $L_{\rho}$, and the momentum mixing length $L_{m}$, both defined by Odier et al. (2009) as

$$L_{m}\equiv\left(\frac{P}{S^{3}}\right)^{1/2},\ L_{\rho}\equiv\left(\frac{B}{N^{2}S}\right)^{1/2}\;\text{ ,}$$

(34)

are coupled via the turbulent Prandtl number ${Pr_{t}}$, which here is very close to one, since

$$L_{\rho}=L_{m}\left(\frac{BS^{2}}{PN^{2}}\right)^{1/2}=L_{m}\left(\frac{{Rf}}{Ri}\right)^{1/2}\simeq{Pr_{t}}^{-1/2}L_{m}\;\text{ .}$$

(35)

Furthermore, they coincide with the Corrsin length scale by the stationarity constraint from (31). The expected scalings $L_{\rho}\approx L_{C}$ and $L_{\rho}\approx L_{m}$ are both clearly apparent in figure 6a across a wide range of Reynolds numbers.

The decrease of $L_{OC}$ implies that the total dynamic-range associated with the scalar anomalously scales with the shear Reynolds number ${Re_{s}}$. We define the total range of dynamic scales for turbulence and the scalar as

	$\displaystyle\Delta\ell_{t}$	$\displaystyle=(L_{L}-L_{K})\;\text{ and}$		(36)
	$\displaystyle\Delta\ell_{\rho}$	$\displaystyle=(L_{OC}-L_{B})\;\text{ ,}$		(37)

respectively, and show them as a function of Reynolds number in figure 6b. With respect to the turbulence scales, after an initial small ${Re_{s}}$ transient, the range of total dynamic scales appears to increase with ${Re_{s}}^{4/3}$, as predicted by our definitions in §2.1.1. For the scalar, on the other hand, there is clearly a flatter-than-predicted scaling regime at high Reynolds number which is characteristic of the transient outer scales observed in figure 6a. Furthermore, the range of dynamics-scales associated with the scalar is significantly smaller than that of the turbulence, suggesting that if a dynamic-range threshold exists, it will not occur simultaneously for the scalar and the turbulence.

In the previous section, we have presented evidence that the ratio of the potential energy spectra to a longitudinal energy spectra appears to be consistent with the underlying scaling arguments central to coexistent Kolmogorov and Obukhov-Corrsin regimes. The value of the key parameter $\Pi$ can be expressed in terms of the one-dimensional universal constants, at high Reynolds number, by

$$\Pi=\frac{\beta}{C}=\frac{6}{11}\frac{\beta_{1}}{C_{1}}.$$

(45)

For these flows, the parameter $\Pi$ should be assumed to be universal to the extent that $C$ and $\beta$ are. For reasons outlined in Sreenivasan (1991) and also observed here, measuring individual constants from a sheared flow is fundamentally problematic and furthermore measuring $\beta$ requires an accurate measurement of $\chi$, which is inherently difficult in experimental flows.

Nonetheless, estimates of the (3D) Kolmogorov and Obukhov-Corrsin constants in the literature would imply that $\Pi\approx 0.42$ (Wyngaard & Coté, 1971; Sreenivasan, 1991; Sreenivasan & Kailasnath, 1996; Yeung, 2002; de Bruyn Kops, 2015) with estimates of the individual constants typically being reported within approximately 15% of each other in the literature cited. Robust estimates of $\Pi$ from the relation (45) and empirical observations in figure 7 indicate $\Pi\approx 0.39$ which is certainly within the range of values reported in literature. We demonstrate that the important expression (45) appears to be valid within the range of uncertainty for reported values of $\beta/C$ in figure 9. We also remark that the range of $\Pi$ is small with respect to values of $\beta/C$ for the Reynolds number space considered. Such an apparently limited range of $\Pi$ has also been reported by Venayagamoorthy & Stretch (2006).

Furthermore, given the empirical observation that $L_{OC}=L_{O}$ (figure 6a) at sufficiently high Reynolds number,

$$\Pi\sim{Fr}.$$

(46)

Therefore, the stationary Froude number can be thought of as a simple consequence of the two universal constants at high Reynolds numbers, as validated in figure 9, which further implies $N_{\ast}\approx 1$, remembering (21).

This finally suggests that the behaviour at lower Reynolds number may be parameterised by a function $f_{\Pi}$, i.e. at unity Prandtl number

$$\Pi=\frac{N_{\ast}}{{S_{\ast}}\sqrt{{Ri}}}\approx{Fr}f_{\Pi}({Re_{s}}).$$

(47)