4.1. Surface conditional statistics

The analytical section presented above reveals that the displacement speed $S_d$ contains many of the key physics in the behaviour of iso-scalar surfaces. Depending on the situation, it may incorporate different processes such as diffusion, chemical reactions or any other source terms acting on the scalar field. It is thus important to understand how $S_d$ evolves along the iso-surface when $R_\lambda$ and/or $Sc$ are varied. In our situation, the displacement speed contains up to three components, one due to diffusion in the normal direction noted $S_d^n$, another due to diffusion in the tangential direction $S_d^c = 2D\mathcal {H}$ and a last contribution associated with the imposed mean scalar gradient noted $S_d^s$. Their respective expressions are summarized in table 2.

Table 2. Expression of the different components of the displacement speed in forced turbulence with a mean scalar gradient $G_\xi$.

As it is generally done in reacting flows (Gran et al. Reference Gran, Echekki and Chen1996; Peters et al. Reference Peters, Terhoeven, Chen and Echekki1998), the variations of $S_d$ along the surface $\xi (\boldsymbol {x},t)=\xi _0$ are analysed through the surface weighted average of its different components conditioned on the mean curvature $\mathcal {H}$. These surface conditioned statistics are noted $\langle S_d^i|\mathcal {H}\rangle _s$ ($i = \lbrace c, n, s\rbrace$ denotes the contribution to the displacement speed). Such conditional statistics are relevant notably for providing insights into the curvature component of the stretch rate $\mathbb {K}_C$, which reads as the mean product of the displacement speed $S_d$ by the mean curvature $\mathcal {H}$.

We first analyse the effect of $R_\lambda$ on $S_d$ and its components. Surface conditioned statistics of $S_d$, $S_d^d$ and $S_d^c$ are portrayed in figure 3(a), while those of $S_d^n$ and $S_d^s$ are displayed in figure 3(b). We consider here an iso-value of $\xi _0=0$. Note that in figure 3(a) the ordinate axis is ten times larger than the one in figure 3(b), which shows that the source term and normal diffusion contributions of $S_d$ are much smaller than the tangential diffusion component. Therefore, assuming $S_d \approx 2D\mathcal {H}$ appears as a reasonable assumption irrespective of the Reynolds number. In figure 3(b) we also note that variations of $S_d^s$ decrease in amplitude when $R_\lambda$ increases. This suggests that the contribution due to the mean scalar gradient might become negligible at sufficiently large $R_\lambda$. It is worth noting that the normal diffusion component of the displacement speed $S_d^n$ reveals a non-monotonic evolution with respect to the mean curvature $\mathcal {H}$. When the surface is only slightly curved ($|\mathcal {H}|$ is small), $S_d^n$ is negative (positive) for positive (negative) values of $\mathcal {H}$, while the contrary is observed for highly curved regions. The variations of $S_d^n$ with the mean curvature does not depend on the Reynolds number for small values of $\mathcal {H}$.

Figure 3. Surface averaged displacement speed and its components conditioned on the mean curvature $\mathcal {H}$ for $\xi _0=0$. From dark to light, case F0 ($Sc=1.0$) to F4. In (a) $S_d$ (full lines), $S_d^d$ (dashed lines) and $S_d^c$ (dotted lines) are portrayed while $S_d^n$ (dashed lines) and $S_d^s$ (full lines) are displayed in (b).

The results for different Schmidt numbers are displayed in figure 4. One notes again that the tangential diffusion component of the displacement speed $S_d^c$ is predominant, thereby emphasizing the role played by curvature. All curves collapse relatively well when the mean curvature is scaled by the scalar diffusivity $D$. When the Schmidt number decreases, the contribution due to the source term $S_d^s$ increases and so does the normal diffusion contribution $S_d^n$.

Figure 4. Surface averaged displacement speed and its components conditioned by the mean curvature $\mathcal {H}$ for $\xi _0=0$. From dark to light, case F0, $Sc=1.0$ to $Sc=0.1$. In (a) $S_d$ (full lines), $S_d^d$ (dashed lines) and $S_d^c$ (dotted lines) are portrayed while $S_d^n$ (dashed lines) and $S_d^s$ (full lines) are displayed in (b).

The different components of the displacement speed $\langle S_d^i \,|\, \mathcal {H} \rangle _s$ for a different iso-scalar value $\xi _0 = \xi _{rms}$ are plotted in figures 5 and 6, respectively. We note here again that $S_d^s$ and $S_d^n$ are about 3 to 10 times smaller in amplitude than $S_d^d$ and $S_d^c$. In contrast with $\xi _0 = 0$, $S_d^n$ appears positive for all values of curvature and irrespective of the Schmidt and Reynolds numbers. Although the trends are quantitatively different from what was observed for ${\xi _0 = 0}$, we note that $S_d$ remains strongly dependent to $\mathcal {H}$. In addition, a careful examination of figure 5 reveals that when $R_\lambda$ increases, the contribution of $S_d^s$ and $S_d^n$ decreases which leads to a better correlation between $S_d$ and $2D\mathcal {H}$. We can speculate that at asymptotically large $R_\lambda$, the assumption $S_d \approx 2D\mathcal {H}$ holds true irrespective of the iso-scalar value. At constant $R_\lambda$, the contribution of $S_d^s$ and $S_d^n$ increases when the scalar diffusivity increases (figure 6). This suggests that the different contributions to $S_d$ for a different iso-level $\xi _0$ should be better studied in terms of the Péclet number. In this context, we expect $S_d \approx 2D \mathcal {H}$ to hold with better accuracy when the Péclet number increases.

Figure 5. Surface averaged displacement speed and its components conditioned on the mean curvature $\mathcal {H}$ for $\xi _0=\xi _{rms}$. From dark to light, case F0 ($Sc=1.0$) to F4. In (a) $S_d$ (full lines), $S_d^d$ (dashed lines) and $S_d^c$ (dotted lines) are portrayed while $S_d^n$ (dashed lines) and $S_d^s$ (full lines) are displayed in (b).

Figure 6. Surface averaged displacement speed and its components conditioned by the mean curvature $\mathcal {H}$ for $\xi _0=\xi _{rms}$. From dark to light, case F0, $Sc=1.0$ to $Sc=0.1$. In (a) $S_d$ (full lines), $S_d^d$ (dashed lines) and $S_d^c$ (dotted lines) are portrayed while $S_d^n$ (dashed lines) and $S_d^s$ (full lines) are displayed in (b).

It is worth finally stressing that our conclusion that the tangential diffusion dominates over the other components of the displacement speed, is built upon the observations of surface weighted averaged values conditioned by the mean curvature $\mathcal {H}$. This observable cannot reveal how $S_d$ and its components are statistically distributed for a given $\mathcal {H}$. There is nothing precluding that locally, for a given value of mean curvature, the normal diffusion and/or the source term contributions to $S_d$ dominate over the tangential diffusion. This is likely to be observed for low amplitudes of $\mathcal {H}$ where the probability density function of $\mathcal {H}$ is generally concentrated. More insights into this aspect could be provided by studying the joint probability density function of $S_d^i$ and $\mathcal {H}$. This is far beyond the scope of the present work and is left for future investigations.

4.2. Second-order structure functions

The present framework is first invoked to explore the dynamics of iso-scalars with an imposed mean scalar gradient in stochastically forced turbulence. In this situation both the flow and scalar characteristics are statistically stationary and the displacement speed has two components, one arising from the diffusive term $S_d^d$, and one due to the imposed mean gradient $S_d^s$. We focus on the particular effect of the Reynolds and Schmidt numbers.

The Reynolds number dependence of $\langle (\delta \phi )^2\rangle _{\mathbb {R}, \mathbb {E}, \varOmega }$ for different values of the iso-scalar is first considered. Results are presented in figure 7 for $R_\lambda$ ranging from 88 to 530 and for two values of the iso-scalar, i.e. $\xi _0 = 0$ (figure 7a) and $\xi _0 = \xi _{rms}$ (figure 7b). Here $\langle (\delta \phi )^2\rangle _{\mathbb {R}, \mathbb {E}, \varOmega }$ is normalized by its asymptotic value at large scales, i.e. $2 \langle \phi \rangle _{\mathbb {E}, \mathbb {R}} (1 - \langle \phi \rangle _{\mathbb {E}, \mathbb {R}})$ while the separation $r$ is normalized by $L_\varSigma = 4\langle \phi \rangle _{\mathbb {E}, \mathbb {R}} (1 - \langle \phi \rangle _{\mathbb {E}, \mathbb {R}})/\varSigma$. Using the normalization with $L_\varSigma = 4\langle \phi \rangle _{\mathbb {E}, \mathbb {R}} (1 - \langle \phi \rangle _{\mathbb {E}, \mathbb {R}})/\varSigma$, the asymptotic limits at large and small scales intersect at $r/L_\varSigma = 1$. Our definition for $L_\varSigma$ finds its inspiration in Lebas et al. (Reference Lebas, Menard, Beau, Berlemont and Demoulin2009) and Thiesset et al. (Reference Thiesset, Duret, Ménard, Dumouchel, Reveillon and Demoulin2020). A somehow similar definition for $L_\varSigma$ was conjectured by Peters (Reference Peters1992) for premixed flames. In the context of the Bray–Moss–Libby model (Bray & Moss Reference Bray and Moss1977; Libby & Bray Reference Libby and Bray1980), $L_\varSigma$ is known as the wrinkling scale (Kulkarni et al. Reference Kulkarni, Buttay, Kasbaoui, Attili and Bisetti2021), although the surface density is defined differently by the latter authors.

Figure 7. Evolution of $\langle (\delta \phi )^2\rangle _{\mathbb {E}, \mathbb {R}, \varOmega } / 2 \langle \phi \rangle _{\mathbb {E}, \mathbb {R}} (1 - \langle \phi \rangle _{\mathbb {E}, \mathbb {R}})$ with increasing $R_\lambda$. The separation $r$ is normalized by $L_\varSigma = 4 \langle \phi \rangle _{\mathbb {E}, \mathbb {R}} (1 - \langle \phi \rangle _{\mathbb {E}, \mathbb {R}}) / \varSigma$. The local scaling exponent is also plotted in the inset. The dotted grey lines represent the asymptotic theoretical limits at large and small scales. Results are shown for (a) $\xi _0 =0$, (b) $\xi _0 = \xi _{rms}$.

One observes that all curves collapse at small scales where $\langle (\delta \phi )^2\rangle _{\mathbb {R}, \mathbb {E}, \varOmega } \to \varSigma r / 2$ and at large scales where $\langle (\delta \phi )^2\rangle _{\mathbb {R}, \mathbb {E}, \varOmega } \to 2 \langle \phi \rangle _{\mathbb {E}, \mathbb {R}} (1 - \langle \phi \rangle _{\mathbb {E}, \mathbb {R}})$. The most expert readers will probably notice that this behaviour is also observed when two-point statistics of the velocity (or scalar) field are normalized by the Taylor (or Corrsin) microscale (Thiesset et al. Reference Thiesset, Schaeffer, Djenidi and Antonia2014). Speculatively, this indicates that $L_\varSigma$ plays for $\langle (\delta \phi )^2\rangle _{\mathbb {R}, \mathbb {E}, \varOmega }$ the same role as the Taylor (Corrsin) microscale for normalizing the two-point statistics of the velocity (scalar) field.

At intermediate scales, the influence of the Reynolds number is perceptible. It is observed that a pseudo ‘inertial range’ is forming whose extent increases with $R_\lambda$. The local scaling exponent $\partial \log (\langle (\delta \phi )^2\rangle _{\mathbb {R}, \mathbb {E}, \varOmega }) / \partial \log (r)$ is plotted in the inset of figure 7 and reveals that a power law with an exponent of about $\zeta \approx 0.36-0.38$ applies over about one decade at $R_\lambda = 530$. As shown by Vassilicos & Hunt (Reference Vassilicos and Hunt1996) and Elsas et al. (Reference Elsas, Szalay and Meneveau2018), the distribution of $\langle (\delta \phi )^2\rangle _{\mathbb {R}, \mathbb {E}}$ at intermediate scales contains information about the fractal characteristics of iso-scalar surfaces. The scaling exponents relate to the fractal dimension of the iso-surface by $D_f= 3-\zeta$. The same relation is used in the community of fractal aggregates (see, e.g. Sorensen Reference Sorensen2001; Morán et al. Reference Morán, Fuentes, Liu and Yon2019). Note that what we call here a fractal dimension should rather be identified as a Kolmogorov capacity (Vassilicos & Hunt Reference Vassilicos and Hunt1991; Vassilicos Reference Vassilicos1992). We also exclude the possibility that the fractal dimension of the intersection of the iso-scalar volume with a line (what we actually measure using two-point statistics of the phase indicator) may be different from the fractal dimension of the iso-volume itself (Vassilicos Reference Vassilicos1992).

The numerical value for $D_f$ is found to be in the range {2.62–2.64}, in quite good agreement with the DNS value reported by Iyer et al. (Reference Iyer, Schumacher, Sreenivasan and Yeung2020) in the exact same numerical configuration (they find $D_f=2.67$ at $R_\lambda = 650$ for $\xi _0 =0$ using the box-counting method) and the theoretical analysis of Mandelbrot (Reference Mandelbrot1975) or Grossmann & Lohse (Reference Grossmann and Lohse1994) (also reproduced by Iyer et al. Reference Iyer, Schumacher, Sreenivasan and Yeung2020) providing a value of $8/3$. When $\xi _0 = \xi _{rms}$, the value for the fractal dimension is roughly the same although Iyer et al. (Reference Iyer, Schumacher, Sreenivasan and Yeung2020) showed that the fractal dimension decreases when the threshold $\xi _0$ is moved away from the mean value $\xi _0 = 0$. It is also worth noting that the scale dependence of the local slope in the range $L_\varSigma \leq r \leq 10L_\varSigma$ does not exceed 5 %. This is in contrast with the results presented by Iyer et al. (Reference Iyer, Schumacher, Sreenivasan and Yeung2020) (see their figure 2) where there is no distinct scaling range for $\xi _0 = 0$. This suggests that, in agreement with the observations of Elsas et al. (Reference Elsas, Szalay and Meneveau2018), measuring the fractal dimension using the second-order structure function of the iso-scalar field is probably more robust than the one inferred from the box-counting method.

The flow under consideration is anisotropic due to the presence of the mean scalar gradient. We here coped with this by employing the partial angular average along three coordinates of $\boldsymbol {r}$. It is thus worth evaluating if the above features for $\langle (\delta \phi )^2\rangle _\mathbb {E,R}$ are retrieved along the different directions of the separation vector $\boldsymbol {r}$. This point is addressed in Appendix F where similar trends are observed, irrespective of the direction. Only some small differences between the directions parallel and perpendicular to the mean scalar gradient appear at the large scales.

In Appendix D we also consider normalizing the separation $r$ by the radius of gyration $R_g$. The latter can be computed directly from $\langle (\delta \phi )^2\rangle _{\mathbb {R}, \mathbb {E}, \varOmega }$ by (Sorensen Reference Sorensen2001; Yon et al. Reference Yon, Morán, Ouf, Mazur and Mitchell2021)

(4.1) \begin{equation} R_g^2 = \frac{1}{2} \frac{\displaystyle\int_0^\infty r^4 A(r)\,{\rm d} r}{\displaystyle\int_0^\infty r^2 A(r)\,{\rm d} r}, \end{equation}

where $A(r)$ is the correlation function normalized in such a way that $A(r)=1$ at $r=0$ and $A(r)=0$ at large scales, viz.

(4.2)\begin{equation} A(r) = 1 - \frac{\langle (\delta \phi)^2\rangle_{\mathbb{R}, \mathbb{E}, \varOmega}}{2 \langle \phi \rangle_{\mathbb{E}, \mathbb{R}} (1 - \langle \phi \rangle_{\mathbb{E}, \mathbb{R}})}. \end{equation}

The results presented in figure 20 of Appendix D show that the radius of gyration is a characteristic scale of the distribution $\langle (\delta \phi )^2\rangle _{\mathbb {R}, \mathbb {E}, \varOmega }$ at large scales. Hence, $R_g$ plays for $\langle (\delta \phi )^2\rangle _{\mathbb {R}, \mathbb {E}}$ the same role as the integral length scale for normalizing the two-point statistics of the velocity (or scalar) field.

In Appendix E we also test the appropriateness of using the standard deviation of mean curvature $\mathcal {H}_{rms}$ as a similarity variable. This type of normalization is expected to hold at small scales. Indeed, going back to (2.17), and further assuming that $\langle \mathcal {G}\rangle _s \ll \langle \mathcal {H}^2\rangle _s$, we have

(4.3)\begin{equation} \frac{\langle (\delta \phi)^2\rangle_{\mathbb{E}, \mathbb{R}, \varOmega} } {\varSigma \mathcal{H}_{rms}^{ -1}} = \frac{1}{2} r \mathcal{H}_{rms} \left(1-\frac{\left(r \mathcal{H}_{rms}\right)^2}{8}\right), \end{equation}

which is thus expected to be independent of Reynolds and Schmidt numbers when plotted in terms of $r\mathcal {H}_{rms}$. The evolution of $\langle (\delta \phi )^2\rangle _{\mathbb {E}, \mathbb {R}, \varOmega }$ for different $R_\lambda$ when the separation $r$ is normalized by $r\mathcal {H}_{rms}$ is presented in figure 24 of Appendix E. It appears that $r\mathcal {H}_{rms}$ plays for the phase indicator field the same role as the Kolmogorov (or Batchelor) length scale for normalizing the two-point statistics of the velocity (scalar) field.

The effect of Schmidt number on $\langle (\delta \phi )^2\rangle _{\mathbb {E}, \mathbb {R}, \varOmega }$ is plotted in figure 8 for $R_\lambda = 88$ and $Sc$ ranging from 0.1 to 1 and $\xi _0 = 0$ (figure 8a) and $\xi _{rms}$ (figure 8b). We observe again that normalizing the separation $r$ by $L_\varSigma$ and $\langle (\delta \phi )^2\rangle _{\mathbb {E}, \mathbb {R}, \varOmega }$ by $2 \langle \phi \rangle _{\mathbb {E}, \mathbb {R}} (1 - \langle \phi \rangle _{\mathbb {E}, \mathbb {R}})$ yields a convincing collapse of the different curves at both small and large scales. The Schmidt number effects are perceptible only at intermediate scales where the scale distribution widens with increasing $Sc$. This means that increasing the diffusivity of the scalar tends to decrease the morphological content of the iso-scalar fields. This is the first evidence that diffusion acts as a restoration effect which counteracts the influence of turbulent straining. More details on this aspect will be given later when examining the budgets of $\langle (\delta \phi )^2\rangle _{\mathbb {E}, \mathbb {R}, \varOmega }$.

Figure 8. Evolution of $\langle (\delta \phi )^2\rangle _{\mathbb {E}, \mathbb {R}, \varOmega } / 2 \langle \phi \rangle _{\mathbb {E}, \mathbb {R}} (1 - \langle \phi \rangle _{\mathbb {E}, \mathbb {R}})$ with increasing $Sc$. The separation $r$ is normalized by $L_\varSigma = 4 \langle \phi \rangle _{\mathbb {E}, \mathbb {R}} (1 - \langle \phi \rangle _{\mathbb {E}, \mathbb {R}}) / \varSigma$. The dotted grey lines represent the asymptotic theoretical limits at large and small scales. Results are shown for (a) $\xi _0 =0$, (b) $\xi _0 = \xi _{rms}$.

4.3. Transfer term

We now address the influence of Reynolds and Schmidt numbers on the transfer term $-\langle \boldsymbol {\nabla _r} \boldsymbol {\cdot } \langle (\delta \boldsymbol {u})(\delta \phi )^2\rangle _{\mathbb {E}, \mathbb {R}} \rangle _{\varOmega }$. In figure 9 we consider the case where $R_\lambda$ is ranging from 88 to 530 and $Sc=1.0$. The transfer term is normalized by $2 \mathbb {K}_T \langle \phi \rangle _{\mathbb {E}, \mathbb {R}} (1 - \langle \phi \rangle _{\mathbb {E}, \mathbb {R}})$ while the separation is divided by $L_\varSigma$. This normalization is found to yield a good collapse of all curves at small scales. The specific evolution of the strain rate $\mathbb {K}_T$ with respect to $R_\lambda$ and $Sc$ will be discussed later. Although visible, the influence of the iso-value $\xi _0$ is rather limited, at least when the latter is moved from $\xi _0 = 0$ to $\xi _0 = \xi _{rms}$.

Figure 9. Evolution of the transfer term $-\langle \boldsymbol {\nabla _r} \boldsymbol {\cdot } \langle (\delta \boldsymbol {u})(\delta \phi )^2\rangle _{\mathbb {E}, \mathbb {R}} \rangle _{\varOmega }$ normalized by $2 \mathbb {K}_T \langle \phi \rangle _{\mathbb {E}, \mathbb {R}} (1 - \langle \phi \rangle _{\mathbb {E}, \mathbb {R}})$ with increasing $R_\lambda$. The separation $r$ is normalized by $L_\varSigma$. The local scaling exponent is also plotted in the inset. The dotted grey lines represent the asymptotic theoretical limits at small scales. Results are shown for (a) $\xi _0=0$, (b) $\xi _0 = \xi _{rms}$. The colour legend is the same as in figure 7.

In figure 21 of Appendix D we also report that the transfer term is independent of $R_\lambda$ in the large-scales limit when the separation is normalized by $R_g$, while $\langle \boldsymbol {\nabla _r} \boldsymbol {\cdot } \langle (\delta \boldsymbol {u})(\delta \phi )^2\rangle _{\mathbb {E}, \mathbb {R}} \rangle _{\varOmega }$ is divided by a sort of turbulent strain felt at a scale $R_g$ which can be written as $\sqrt {\langle k \rangle }/R_g$ ($\langle k \rangle$ is the turbulent kinetic energy). Consequently, $R_g$ and $\sqrt {\langle k \rangle }/R_g$ are appropriate for normalizing the transfer term in the large-scale limit. Figure 25 of Appendix E proves that the small-scale similarity variables for the transfer term are $\mathcal {H}_{rms}^{-1}$ and $\mathbb {K}_T \varSigma \mathcal {H}_{rms}^{-1}$ which plays for $\phi$ the same role as the Kolmogorov (Batchelor) scales for the velocity (scalar) field.

In figure 9 we also show the local scaling exponent of the transfer term. Although the scaling range appears more restricted than the one observed for second-order moments, there seems to be a plateau forming around a value of about $\{ -0.21;-0.23\}$ at the larger $R_\lambda$. Let us naively assume that, at intermediate scales, the flux can be written as

(4.4)\begin{equation} \langle (\delta \boldsymbol{u})(\delta \phi)^2\rangle_{\mathbb{E}, \mathbb{R}} \sim \langle (\delta u_{||})^2\rangle_\mathbb{R,E}^{1/2} \langle (\delta \phi)^2\rangle_\mathbb{R,E} \sim r^{ \zeta_u/2 + \zeta}. \end{equation}

Here, $\delta u_{||} = \delta \boldsymbol {u} \boldsymbol {\cdot } \boldsymbol {r}/|\boldsymbol {r}|$ is the longitudinal increment of velocity. If we further state that $\zeta _u$, the scaling exponent for the velocity structure function, is equal to $\zeta _u = 2/3$, we obtain that the transfer term should scale as $r^{\zeta -2/3}= r^{7/3 - D_f}$ which, for $\zeta =\lbrace 0.36;\ 0.38 \rbrace$, gives a scaling exponent of $\lbrace -0.29;\ -0.31\rbrace$. Our numerical data indicate a value around $\lbrace -0.21; -0.23\rbrace$ for the transfer term scaling exponent which is in reasonable agreement with this crude scaling analysis. If we account for internal intermittency, i.e. $\zeta _u> 2/3$, the predicted exponent of the transfer term is closer to the numerical value. Note that this reasoning holds also in the small-scale limit, where $\langle (\delta u_{||})^2\rangle ^{1/2} \sim \langle (\delta \phi )^2\rangle \sim r^1$ and, hence, the transfer term should scale as $r^1$, which is numerically observed. To give this scaling analysis a bit more strength, we use the closure proposed by de Divitiis (Reference de Divitiis2014, Reference de Divitiis2016, Reference de Divitiis2020) which has the advantage of not relying on a parametrized turbulent diffusion hypothesis. When adapted to $\langle (\delta \phi )^2\rangle _{\mathbb {E}, \mathbb {R}}$, the latter writes as

(4.5)\begin{equation} -\langle \boldsymbol{\nabla_r} \boldsymbol{\cdot} \langle (\delta \boldsymbol{u}) (\delta \phi)^2\rangle_{\mathbb{E}, \mathbb{R}} \rangle_{\varOmega} = \tfrac{1}{2} {\langle (\delta u_{||})^2\rangle^{1/2}_{\mathbb{E}, \mathbb{R}, \varOmega}} \partial_r \langle (\delta \phi)^2 \rangle_{\mathbb{E}, \mathbb{R}, \varOmega}. \end{equation}

Assuming again that $\langle (\delta u_{||})^2\rangle_{\mathbb{E}, \mathbb{R},\varOmega}$ scales as $r^{2}$ at small scales and $r^{2/3}$ at intermediate scales, we obtain that the transfer term should scale as $r^1$ and $r^{\zeta - 2/3} = r^{7/3 - D_f}$ at small and intermediate scales, respectively. This reasoning is in agreement with the numerical data.

While the Kolmogorov four-fifth law and Yaglom four-thirds law are known to provide a $r^0$ scaling for the transfer term of either velocity or scalar in the inertial range, the one pertaining to the phase indicator is substantially different and is proved to relate to the fractal dimension of the iso-surface. According to our elaborations, a fractal dimension of $8/3$ translates into a $r^{-1/3}$ scaling for the transfer term of iso-volumes while a fractal dimension of $7/3$ results in a $r^0$ scaling.

The effect of Schmidt number on the transfer term is displayed in figure 10(a) for $\xi _0 = 0$ and figure 10(b) for $\xi _0 = \xi _{rms}$. In both cases, $R_\lambda = 88$. When $Sc$ increases from 0.1 to 1.0, all curves collapse well at small scales thereby complying with the $\lbrace L_\varSigma, 2 \mathbb {K}_T \langle \phi \rangle _{\mathbb {E}, \mathbb {R}} (1 - \langle \phi \rangle _{\mathbb {E}, \mathbb {R}}) \rbrace$ scaling. It is further observed that decreasing the diffusivity of the scalar field (i.e. increasing the Schmidt number) acts in widening the range of scales over which the transfer term operates. The same trend was observed for increasing Reynolds numbers. This suggests that the appropriate non-dimensional number for characterizing the phase indicator scale distribution and its transfer is likely to be the Péclet number. This assertion will be discussed in more detail later in this paper.

Figure 10. Evolution of the transfer term $-\langle \boldsymbol {\nabla _r} \boldsymbol {\cdot } \langle (\delta \boldsymbol {u})(\delta \phi )^2\rangle _{\mathbb {E}, \mathbb {R}} \rangle _{\varOmega }$ normalized by $2 \mathbb {K}_T \langle \phi \rangle _{\mathbb {E}, \mathbb {R}} (1 - \langle \phi \rangle _{\mathbb {E}, \mathbb {R}})$ with increasing $Sc$. The separation $r$ is normalized by $L_\varSigma$. The dotted grey lines represent the asymptotic theoretical limits at small scales. Results are shown for (a) $\xi _0=0$, (b) $\xi _0 = \xi _{rms}$. The colour legend is the same as in figure 8.

4.4. Two-point budget

The different terms of the angularly averaged budget (2.15) for $Sc=1.0$, $88 \leq R_\lambda \leq 530$ and for two values for the iso-scalar $\xi _0 = 0$ and $\xi _0 = \xi _{rms}$ are presented in figure 11. Here again, the different terms are normalized by $2 \mathbb {K}_T \langle \phi \rangle _{\mathbb {E}, \mathbb {R}} (1 - \langle \phi \rangle _{\mathbb {E}, \mathbb {R}})$ while the separation is divided by $L_\varSigma$. The normalization by the large-scale quantities $R_g$ and $\sqrt {\langle k \rangle }/R_g$ is reported in figure 22 of Appendix D while the one based on $\mathcal {H}_{rms}$ and $\mathbb {K}_T\varSigma /\mathcal {H}_{rms}$ is plotted in figure 26 of Appendix E.

Figure 11. Budget of $\langle (\delta \phi )^2\rangle _{\mathbb {E}, \mathbb {R}, \varOmega }$ with increasing $R_\lambda$. Full lines: transfer term, dashed lines: $S_d^d$-term, dash-dotted lines: $S_d^s$-term. All contributions are normalized by $2 \mathbb {K}_T \langle \phi \rangle _{\mathbb {E}, \mathbb {R}} (1 - \langle \phi \rangle _{\mathbb {E}, \mathbb {R}})$ while the separation $r$ is normalized by $L_\varSigma$. Results are shown for (a) $\xi _0=0$, (b) $\xi _0 = \xi _{rms}$. The colour legend is the same as in figure 7.

Figure 11 reveals that the transfer term is positive which means that, as expected, the action of turbulence stirs, stretches and folds the scalar field thereby increasing its tortuousness and its morphological content. The diffusive component of the interface propagation term, i.e. the term due to $S_d^d$, is negative and thus acts in smoothing the interface. Peters (Reference Peters1992) used to refer to the process associated with $S_d$ as a kinematic restoration effect which appears indeed aptly named as it tends to counteract the influence of turbulent strain by smoothing the interface. The term in the two-point budget associated with the imposed mean gradient, i.e. the term due to $S_d^s$, is negative for an iso-scalar value $\xi _0 = 0$ and positive for $\xi _0 = \xi _{rms}$. This indicates that the imposed mean gradient decreases the morphological content of the iso-scalar close to the $\xi _0 = 0$ iso-value and redistributes it to the scalar iso-values away from the mean.

The influence of $R_\lambda$ is also visible in figure 11. When normalized by $2 \mathbb {K}_T \langle \phi \rangle _{\mathbb {E}, \mathbb {R}} (1 - \langle \phi \rangle _{\mathbb {E}, \mathbb {R}})$, the different terms collapse at small scales and decrease in amplitude in the intermediate range of scales. When $R_\lambda$ increases, the range of scales over which the different terms of the budget are contributing increases. Using the large-scale normalization (see figure 22 in Appendix D), the terms collapse at large scales and their respective amplitude increases with $R_\lambda$. The evolution of the different terms of the budget using the small-scale similarity variables is presented in figure 26 of Appendix E. It is also worth noting that, at small, up to intermediate scales, the relative influence of the $S_d^s$-term compared with, for example, the transfer term decreases when $R_\lambda$ increases. This suggests that in the limit of very large $R_\lambda$ the strain rate and the diffusion terms balance, while the source term due to the imposed mean gradient remains concentrated only at the large scales. In other words, at large $R_\lambda$, the strain rate $\mathbb {K}_T$ and the curvature component of the stretch rate due to $S_d^d$, i.e. $\mathbb {K}_C^d = -2 \langle S_d^d \mathcal {H} \rangle _s$, balance, whilst the curvature component of the stretch rate due to $S_d^s$, i.e. $\mathbb {K}_C^s = -2 \langle S_d^s \mathcal {H} \rangle _s$, tends to zero. For $\xi _0 = 0$, the area weighted averaged displacement speed is zero. Figure 11(a) confirms that all terms tend to zero at large scales in agreement with (2.22). For $\xi _0 = \xi _{rms}$, only the transfer term approaches zero when $r \to \infty$ while the $S_d^d$- and $S_d^s$-terms balance. The limit of the $S_d^d$-term at large scales is also in agreement with (2.22) which is displayed by the horizontal grey dotted lines in figure 11(b). The balance between the $S_d^d$- and $S_d^s$-terms at large scales suggests that the volume of the excursion set which naturally decreases due to diffusion effect is exactly compensated by the imposed mean gradient.

The effect of Schmidt number on the different terms of the budget at constant $R_\lambda =88$ is displayed in figure 12(a) for $\xi _0=0$ and figure 12(b) for $\xi _0=\xi _{rms}$. Here again, the normalization using $L_\varSigma$ and $\mathbb {K}_T$ yields a good collapse of all curves in the limit of small separations. For $\xi _0=0$, decreasing the Schmidt number from 1.0 to 0.1, i.e. increasing the scalar diffusivity by a factor of 10, leads to a smaller amplitude of the diffusion term in the intermediate range of scales. In the same range of scales (i.e. up to $r \approx L_\varSigma$) the transfer term normalized by $\mathbb {K}_T$ appears rather insensitive to Schmidt number variations. The influence of $Sc$ on the transfer term is perceptible only at large scales where it is observed that the scale at which the transfer term approaches zero decreases with increasing scalar diffusivity. The third term in the budget due to the imposed mean gradient is plotted as dash-dotted lines. When $\xi _0=0$, the latter is negative, progressively tends to zero when $Sc$ decreases, and becomes even slightly positive for $Sc=0.1$. This means that when $Sc=0.1$, the scalar is so diffusive that the imposed mean gradient becomes a gain in the budget for this particular iso-value. For $\xi _0 = \xi _{rms}$, the term due to the imposed mean gradient acts at rather large scales, is positive and increases in amplitude when $Sc$ decreases. At large scales the budget is composed of only the diffusion and imposed mean gradient terms, while the transfer term is zero.

Figure 12. Budget of $\langle (\delta \phi )^2\rangle _{\mathbb {E}, \mathbb {R}, \varOmega }$ with increasing $Sc$. Full lines: transfer term, dashed lines: $S_d^d$-term, dash-dotted lines: $S_d^s$-term, dotted lines: $-S_d^d$-term. All contributions are normalized by $2 \mathbb {K}_T \langle \phi \rangle _{\mathbb {E}, \mathbb {R}} (1 - \langle \phi \rangle _{\mathbb {E}, \mathbb {R}})$ while the separation $r$ is normalized by $L_\varSigma$. Results are shown for (a) $\xi _0=0$, (b) $\xi _0 = \xi _{rms}$. The colour legend is the same as in figure 8.

4.5. Strain and curvature components of the stretch rate

We now investigate the evolution of the different normalizing quantities with respect to $R_\lambda$ and $Sc$. When both $R_\lambda$ and $Sc$ vary, it may be more appropriate to define the turbulent Péclet number,

(4.6)\begin{equation} Pe_{\lambda_\xi} = \frac{\sqrt{2 \langle k \rangle /3} \lambda_\xi}{D}. \end{equation}

With this definition, the Péclet number is related to the Schmidt and Reynolds number by

(4.7)\begin{equation} Pe_{\lambda_\xi} = \left(\frac{6R}{10}\right)^{1/2} R_\lambda Sc^{1/2}, \end{equation}

where $R$ is the scalar to mechanical time-scale ratio, i.e.

(4.8)\begin{equation} R = \frac{\langle \xi^2 \rangle}{\langle \epsilon_\xi \rangle} \frac{\langle \epsilon \rangle}{\langle k \rangle}, \end{equation}

which may also vary with $R_\lambda$ and $Sc$.

We first characterize the influence of the Péclet number on $\mathbb {K}_T$, $\mathbb {K}_C^d$ and $\mathbb {K}_C^s$. Here $\mathbb {K}_T$ is inferred from (2.24), i.e. from the slope of the transfer term in the limit of small scales. Similarly, (2.25) reveals that $\mathbb {K}_C^d$ and $\mathbb {K}_C^s$ can be obtained from the slope of the $S_d^d$- and $S_d^s$-terms, respectively, when $r$ approaches zero. Results are portrayed in figure 13 where all quantities are made non-dimensional by multiplying by the Kolmogorov time scale $\tau _\eta = (\nu /\langle \epsilon \rangle )^{1/2}$.

Figure 13. Scaling of the different components of the stretch rate $\mathbb {K} = \mathbb {K}_T + \mathbb {K}_C$ with respect to $R_\lambda$. Results are shown for (a) $\xi _0=0$, (b) $\xi _0 = \xi _{rms}$.

For $Pe_{\lambda _\xi }> 50$, the normalized strain rate $\mathbb {K}_T \tau _\eta$ is nearly constant around a value of about $0.25$. Since this observation holds for both $\xi _0 =0$ and $\xi _0 = \xi _{rms}$, this means that the different iso-scalars experience nearly the same turbulent straining. The value of $0.25\tau _\eta$ is in agreement with the finding of Yeung, Girimaji & Pope (Reference Yeung, Girimaji and Pope1990) who report a value of 0.28 for material surfaces. It is also consistent with the phenomenological model of Thiesset et al. (Reference Thiesset, Maurice, Halter, Mazellier, Chauveau and Gökalp2016b) which gives roughly the same value of 0.28. It is argued by Girimaji & Pope (Reference Girimaji and Pope1992) that the strain rate experienced by propagating surfaces is smaller than that acting on material surfaces since there will be less time for the iso-surface to align with strain. Our value of $0.25$ for $\mathbb {K}_T \tau _\eta$ instead of 0.28 is thus consistent with this argument. In addition, when the scalar diffusivity is increased so that $Pe_{\lambda _\xi } < 50$, we note a substantial decrease of $\mathbb {K}_T \tau _\eta$ which drops down to 0.2. This indicates that the higher the diffusivity, the larger is the displacement speed, and the smaller the time for the iso-surface to align with strain. The constancy of $\mathbb {K}_T \tau _\eta$ at large Péclet numbers is also predicted by the closure of de Divitiis (Reference de Divitiis2014, Reference de Divitiis2016, Reference de Divitiis2020) given by (4.5). Indeed, Kolmogorov's first similarity hypothesis implies that, for $r \to 0$,

(4.9)\begin{equation} \langle (\delta u_{||})^2 \rangle_{\mathbb{E,R},\varOmega} \sim \frac{\langle \epsilon \rangle}{\nu} r^2, \end{equation}

which by virtue of (2.17) and (2.24) gives $\mathbb {K}_T \tau _\eta = {\rm const.}$.

A careful analysis of figure 13 further shows that $K_C^d \tau _\eta$ is always negative and, thus, counteracts the effect of turbulent straining. It also approaches a constant value when the Péclet number increases, but at a smaller rate than $\mathbb {K}_T$. The influence of the imposed mean gradient on the source component of the stretch rate $K_C^s \tau _\eta$ is perceptible at finite Péclet number and it is observed to be negative for $\xi _0 =0$ and positive for $\xi _0 = \xi _{rms}$. Here again, this suggests that the imposed mean gradient acts in redistributing the surface density from the mean iso-value $\xi _0=0$ to iso-values away from the mean. We also note that $K_C^s \tau _\eta$ approaches zero at the highest Péclet number. This indicates that in the limit of very high $Pe_{\lambda _\xi }$, the imposed mean gradient does not influence the evolution of the iso-scalar surface density, the latter being driven only by diffusion and straining effects.

4.6. Characteristic length scales

The previous analysis of two-point statistics highlighted the existence of three characteristic length scales for the excursion set $\phi$. The first one, $L_\varSigma$, is relevant for normalizing the two-point statistics at both small and large scales. The second, $R_g$, applies in the large-scale limit while the third, $\mathcal {H}_{rms}^{-1}$, is relevant at small up to intermediate scales. Some speculations about the connection between $L_\varSigma$ ($\mathcal {H}_{rms}^{-1}$) and the Corrsin (Batchelor) microscales have already been stated earlier in this paper. We now provide more rigorous evidence for this.

First, it is worth recalling that, for isotropic media, the surface density $\varSigma$ is related to the number of zero crossings of the field $\xi (\boldsymbol {x})-\xi _0$ (Torquato Reference Torquato2002). On the other hand, there exist numerous studies that highlight a close relation between the Taylor microscale and the number of zero crossings of turbulent signals (see e.g. Liepmann (Reference Liepmann1949), Sreenivasan, Prabhu & Narasimha (Reference Sreenivasan, Prabhu and Narasimha1983) and Mazellier & Vassilicos (Reference Mazellier and Vassilicos2008), among others). Hence, there are reasons to expect that $L_\varSigma$ and the Corrsin microscale $\lambda _\xi$ are intimately linked. Recall that the Corrsin microscale is here defined by $\lambda _\xi ^2 = 6D\langle \xi ^2\rangle _\mathbb {E,R} / \langle \epsilon _\xi \rangle _\mathbb {E,R}$. Second, given that $R_g$ is the relevant normalizing scale in the large-scale limit, it seems natural to associate $R_g$ with the integral length scale of the scalar field noted $L_\xi$. Here, $L_\xi$ is computed from the scalar fluctuations spectrum $E_\xi (\kappa )$, i.e.

(4.10)\begin{equation} L_\xi = \frac{\rm \pi}{2} \frac{\displaystyle\int_0^\infty \kappa^{{-}1} E_\xi (\kappa)\,{\rm d}\kappa}{\displaystyle\int_0^\infty E_\xi(\kappa)\,{\rm d}\kappa}. \end{equation}

These arguments are tested against numerical data in figure 14(a). While $Pe_{\lambda _\xi }$ is multiplied by roughly 20, the ratio $R_g/L_\xi$ and $\lambda _\xi / L_\varSigma$ remain close to 0.5 with some small departures which do not exceed $\pm 20\,\%$. Hence, it appears that $L_\varSigma \sim \lambda _\xi$ and $R_g \sim L_\xi$ are good approximations. The small departures in the scaling between $R_g$ and $L_\xi$ can be due to several effects. First, some confinement due to the finite ratio between $L_\xi$ and the simulation box size can be at play. Second, the scaling between $R_g$ and $L_\xi$ can also be altered by some finite Péclet number effects. The latter are likely to reveal themselves in the evolution of $R_g/L_\xi$ which is first decreasing before reaching a plateau for Péclet numbers above 50. Such finite Péclet number effects were also noticed by Shete & de Bruyn Kops (Reference Shete and de Bruyn Kops2020). As far as the scaling between $L_\varSigma$ and $\lambda _\xi$ is concerned, it is first worth recalling that the Rice's theorem is valid only if both $\xi$ and $\boldsymbol {\nabla } \xi$ have Gaussian probability density functions and are statistically uncorrelated. While $\xi$ is normally distributed, internal intermittency leads to significant departure from Gaussian distributions for the scalar gradient. At finite Reynolds or Péclet numbers, the assumption of statistical independence between $\xi$ and its derivatives is not likely to hold. As a consequence, all data from the literature (e.g. Sreenivasan et al. Reference Sreenivasan, Prabhu and Narasimha1983) indicate that the Rice theorem is a good approximation for turbulent signals verified within 20 %. Departures are thus of the same magnitude here. Finally, we explored here only two scalar iso-values. It is not excluded that the proportionality between $L_\varSigma$ and $\lambda _\xi$ ceases to apply for some higher iso-values of $\xi _0$.

Figure 14. (a) Ratio of the length scales $L_\xi /R_g$, $\lambda _\xi /L_\varSigma$ and $\mathcal {H}_{rms} \eta _B$, with respect to $Pe_{\lambda _\xi }$ for both $\xi _0=0$ and $\xi _0 = \xi _{rms}$. The dotted lines represent a ratio of 0.5 and 0.22. (b) Surface density $\varSigma$ vs $R_g \mathcal {H}_{rms}$. The lines represent the expectations using a fractal dimension $D_f = 2.62$.

Let us now focus on the scaling of $\mathcal {H}_{rms}$. It is first worth recalling that the relation $\mathbb {K}_T \approx \mathbb {K}_C^d \sim \tau _\eta$ holds relatively well, except maybe at the smallest Péclet numbers (figure 13). On the other hand, the analysis performed in § 4.1 indicates that $S_d \approx 2 D \mathcal {H}$ is a rather safe approximation. With these relations, one can easily conclude that the standard deviation of the mean curvature is proportional to the inverse of the Batchelor length scale $\eta _B$. This result is tested with success in figure 14(a) which proves that the standard deviation of mean curvature is indeed proportional to the Batchelor length scale. (Due to computational limitations, we were unable to estimate $\mathcal {H}_{rms}$ for case F5. Hereafter, we will assume that $\mathcal {H}_{rms} = 0.22/\eta _B$ for this case.) It is worth noting that DNS data give $\eta _B \mathcal {H}_{rms} = {\rm const.}$ even at low Péclet numbers where the approximation $\mathbb {K}_T \approx \mathbb {K}_C^d \sim \tau _\eta$ does not hold anymore. We thus believe that the relation between the Batchelor length scale and the standard deviation of mean curvature is likely to be a general result which does not necessarily requires $\mathbb {K}_T \approx \mathbb {K}_C^d \sim \tau _\eta$. We have also tested the scaling of $\mathcal {H}_{rms}$ with the Obukhov–Corrsin length scale $\eta _{OC} = (D^3/\langle \epsilon \rangle )^{1/4} = \eta / Sc^{3/4}$ (not shown), but the latter was found to be inappropriate. This conclusion is consistent with the findings of, for example, Antonia & Orlandi (Reference Antonia and Orlandi2003) and Donzis, Sreenivasan & Yeung (Reference Donzis, Sreenivasan and Yeung2005) which prove that the Batchelor length scale is the appropriate normalizing the scalar fluctuations scale distributions irrespective of the Schmidt number.

To conclude, for the values of $\xi _0$ investigated here, the geometrical characteristic scales $R_g$, $L_\varSigma$ and $\mathcal {H}_{rms}^{-1}$ of the iso-scalar fields can be related to the somehow more ’usual’ characteristic scales of the scalar field: the integral, Corrsin and Batchelor length scales, respectively. There thus exists an intimate connection between the geometrical and hydrodynamic characteristic scales.

In the appendix we plot the local scaling exponent of $\langle (\delta \phi )^2\rangle _{\mathbb {R,E},\varOmega }$ when the separation is made non-dimensional using either $R_g$ (figure 20) or $\mathcal {H}_{rms}^{-1}$ (figure 24). We observe in figure 24 that the onset of the fractal scaling range starts at a scale proportional to $\mathcal {H}_{rms}^{-1}$. On the other side (figure 20), the end of the scaling range appears at a scale proportional to $R_g$. Therefore, the inner cutoff of the fractal range is related to $\mathcal {H}_{rms}^{-1}$ (and, hence, $\eta _B$) while the outer cutoff is given by $R_g$ (and, hence, $L_\xi$). Note that Sreenivasan et al. (Reference Sreenivasan, Ramshankar and Meneveau1989) conjectured that, for a small Schmidt number, the inner cutoff should be related to the Obukhov–Corrsin length scale $\eta _{OC}$. Our findings indicate that the inner cutoff should better be scaled with the Batchelor length scale.

A fractal scaling should result in (Sreenivasan et al. Reference Sreenivasan, Ramshankar and Meneveau1989)

(4.11)\begin{equation} L_{box}\varSigma = k_f \left(\frac{R_g}{\mathcal{H}_{rms}^{{-}1}}\right)^{D_f - 2}, \end{equation}

where $k_f$ is the fractal prefactor and $L_{box} = 2{\rm \pi}$ is used for normalization as in Shete & de Bruyn Kops (Reference Shete and de Bruyn Kops2020). Figure 14(b) portrays the evolution of surface density $\varSigma$ with respect to $R_g\mathcal {H}_{rms}$ for all Péclet numbers. The log-log representation clearly indicates that a power law is at play with an exponent in very close agreement with $D_f = 2.62$ and $D_f = 2.64$ for $\xi _0 = 0$ and $\xi _0 = \xi _{rms}$, respectively. Surprisingly, (4.11) holds even at the lowest Péclet numbers, although there is no clear scaling range for $\langle (\delta \phi )^2\rangle _{\mathbb {R,E},\varOmega }$. We also note that the fractal prefactor $k_f$ in (4.11) depends in particular on the choice of the iso-scalar but not on the Péclet number.

Expressing $R_g$ and $\mathcal {H}_{rms}$ in terms of $L_\xi$ and $\eta _B$ in (4.11) yields

(4.12)\begin{equation} \varSigma \sim \left(C_\epsilon \frac{L_\xi}{L_t}Pe_\lambda^{{3}/{2}} Sc^{{-}1}\right)^{D_f-2}, \end{equation}

where $C_\epsilon$ is the kinetic energy dissipation constant and $Pe_\lambda$ is the Taylor based Péclet number $Pe_\lambda = R_\lambda Sc$. When expressed in terms of $Pe_{\lambda _\xi }$, (4.12) writes as

(4.13)\begin{equation} \varSigma \sim \left(C_\epsilon \frac{L_\xi}{L_t} R^{{-}3/4 } Pe_{\lambda_\xi}^{{3}/{2}} Sc^{{-}1/4}\right)^{D_f-2}. \end{equation}

Equation (4.12) indicates that, for $Sc=1$, and further omitting the dependence of $C_\epsilon$ and $L_\xi / L_t$ to $R_\lambda$, the surface density of iso-scalars $\varSigma$ should grow as $Pe_\lambda ^{({3}/{2})(D_f-2)} = Pe_{\lambda _\xi }^{({3}/{2})(D_f-2)} = R_\lambda ^{({3}/{2})(D_f-2)}$. If $D_f = 8/3$, we have $\varSigma \sim Pe_\lambda \sim Pe_{\lambda _\xi } \sim R_\lambda$ while $D_f = 7/3$ leads to $\varSigma \sim Pe_\lambda ^{1/2} \sim Pe_{\lambda _\xi }^{1/2} \sim R_\lambda ^{1/2}$. The $Pe_\lambda$ scaling derived in (4.12) is different from the one observed by Shete & de Bruyn Kops (Reference Shete and de Bruyn Kops2020) in a configuration similar to the present one (although the velocity forcing and dealiasing procedures were different). They obtained that $\varSigma \sim Pe_\lambda ^{1/2}$ over an impressive range of Péclet numbers. This result was obtained by averaging the surface areas over 20 different iso-levels covering the range of scalar fluctuations. Since the fractal dimension $D_f$ is known to vary for different iso-levels (Iyer et al. Reference Iyer, Schumacher, Sreenivasan and Yeung2020), lumping together the values for 20 iso-surface areas could have resulted in a different scaling with respect to the Péclet number. Further investigations are required to confirm whether or not (4.12) holds true for the data of Shete & de Bruyn Kops (Reference Shete and de Bruyn Kops2020).

4.7. Decaying turbulence

We now proceed to the analysis of the scale-by-scale budgets in decaying turbulence. In this situation, the time derivative term in (2.15) contributes to the budget. We consider two cases where the production term associated with the imposed mean scalar gradient is either retained or deactivated. For each situation, a Schmidt number of 1.0 and 0.2 is analysed.

We start with the case where the kinetic energy is freely decaying but the imposed mean scalar gradient is maintained. Therefore, the time derivative term in (2.15) is not zero but the displacement speed $S_d$ has a source term contribution $S_d^s$. The different terms of the scale-by-scale budgets are presented in figure 15 for the two Schmidt number values and for three different values of $\xi _0$ from 0 to $\xi _{rms}$. We observe that when normalized by $2 \mathbb {K}_T \langle \phi \rangle _{\mathbb {E}, \mathbb {R}} (1 - \langle \phi \rangle _{\mathbb {E}, \mathbb {R}})$ and $L_\varSigma$, the transfer term is almost independent of the choice of the iso-scalar value. This means that the different iso-surfaces experience the same scale-dependent turbulent straining. The Schmidt number variations are similar to those already documented in the previous section. The contribution due to scalar diffusion is always negative which means that the restoration effect acts in counteracting the turbulent straining. At large scales, increasing $\xi _0$ and decreasing $Sc$ is followed by a increasing amplitude of the $S_d^d$-term which is consistent with an increase of $S_d$ and $D$. The term associated with the imposed mean scalar gradient is positive, acts at larger scales and its amplitude increases with $\xi _0$. The balance between all these terms yields the time variations of $\langle (\delta \phi )^2\rangle _{\mathbb {E}, \mathbb {R}\varOmega }$. This term has a rather small amplitude compared with the three others. We note however that at large scales and for $\xi _0> 0$, the time variations of $\langle (\delta \phi )^2\rangle _{\mathbb {E}, \mathbb {R}\varOmega }$ are positive meaning that the volume $\langle \phi \rangle _{\mathbb {E}, \mathbb {R}}$ tends to increase with time. This is consistent with the observation that in decaying turbulence in the presence of an imposed mean scalar gradient, the variance of $\xi$ grows in time. We also note that at large scales all terms of the budget tend to zero when $\xi _0 = 0$, meaning that the volume $\langle \phi \rangle _{\mathbb {E}, \mathbb {R}}$ for $\xi _0=0$ is conserved.

Figure 15. Budget of $\langle (\delta \phi )^2\rangle _{\mathbb {E}, \mathbb {R}\varOmega }$ with increasing $\xi _0$ in decaying turbulence with imposed mean scalar gradient. Full lines: transfer term, dashed lines: $S_d^d$-term, dash-dotted lines: $S_d^s$-term, dotted lines: $dt$-term. All contributions are normalized by $2 \mathbb {K}_T \langle \phi \rangle _{\mathbb {E}, \mathbb {R}} (1 - \langle \phi \rangle _{\mathbb {E}, \mathbb {R}})$ while the separation $r$ is normalized by $L_\varSigma$. Results are shown for (a) $Sc = 1$, (b) $Sc = 0.2$. Three values of $\xi _0 = 0, 0.5\xi _{rms}, \xi _{rms}$ are displayed from dark to light.

Deactivating the source term leads to $S_d^s = 0$. In this situation the time variations of $\langle (\delta \phi )^2\rangle _{\mathbb {E}, \mathbb {R}, \varOmega }$ are due to the unbalance between the transfer term and diffusion term. Results for $Sc=1.0$ and $0.2$ and for $\xi _0 = 0, 0.5\xi _{rms}$ and $\xi _{rms}$ are presented in figure 16. Here again, we observe that the particular choice of $\xi _0$ does not change drastically the transfer term when the latter is scaled in terms of $2 \mathbb {K}_T \langle \phi \rangle _{\mathbb {E}, \mathbb {R}} (1 - \langle \phi \rangle _{\mathbb {E}, \mathbb {R}})$. Influence of $\xi _0$ and $Sc$ are much more perceptible on the diffusion term and consequently the time derivative term. There is a systematic decrease of the $S_d^d$-term at both intermediate and large scales when $\xi _0$ increases from $0$ to $\xi _{rms}$. We also observe that the slope of the time derivative term at small scales is negative meaning that the surface density is decreasing in time. At large scales, the unsteady term is negative, asymptotes the diffusion term and increases in amplitude with $\xi _0$ and $Sc$.

Figure 16. Budget of $\langle (\delta \phi )^2\rangle _{\mathbb {E}, \mathbb {R}, \varOmega }$ with increasing $\xi _0$ in decaying turbulence without imposed mean scalar gradient. Full lines: transfer term, dashed lines: $S_d^d$-term, dash-dotted lines: $S_d^s$-term, dotted lines: $dt$-term. All contributions are normalized by $2 \mathbb {K}_T \langle \phi \rangle _{\mathbb {E}, \mathbb {R}} (1 - \langle \phi \rangle _{\mathbb {E}, \mathbb {R}})$ while the separation $r$ is normalized by $L_\varSigma$. Results are shown for (a) $Sc = 1$, (b) $Sc = 0.2$. Three values of $\xi _0 = 0, 0.5\xi _{rms}, \xi _{rms}$ are displayed from dark to light.

In summary, forced and decaying turbulence share some of the following common behaviours.

1. (i) The transfer term is positive in both case, meaning that turbulent straining acts in increasing the morphological content of iso-scalar sets.

2. (ii) The contribution due to diffusion (the $S_d^d$-term) is always negative. It thus plays a restoration effect that counteracts the effect of turbulent straining.

3. (iii) Increasing scalar diffusivity (decreasing $Sc$) always leads to a decrease of the iso-scalar morphological content. This is mainly due to the above-mentioned restoration effect, where the latter increases with $D$, which is consistent with the assumption $S_d^d \approx 2D\mathcal {H}$.

In contrast, there are key differences between forced and isotropic turbulence which are summarized below.

1. (i) In forced stationary turbulence, the time derivative term in (2.15) is by definition zero for all scales.

2. (ii) In decaying situations, depending on the iso-scalar value and the presence or absence of a mean scalar gradient, the time derivative term can be either positive or negative as detailed below.

   1. (a) In decaying turbulence, in the absence of a mean scalar gradient, the time derivative term is systematically negative, meaning that both the surface density and iso-scalar volume are decreasing.

   2. (b) In the presence of a mean scalar gradient, the time derivative term can be either positive or negative depending on the iso-scalar value.