Log-Correlated Large-Deviation Statistics Governing Huygens Fronts in Turbulence

Abstract

Analyses have disagreed on whether the velocity $u_T$ of bulk advancement of a Huygens front in turbulence vanishes or remains finite in the limit of vanishing local front propagation speed $u_0$. Here, a connection to the large-deviation statistics of log-correlated random processes enables a definitive determination of the correct small-$u_0$ asymptotics. This result reconciles several theoretical and phenomenological perspectives with the conclusion that $u_T$ remains finite for vanishing $u_0$, which implies a propagation anomaly akin to the energy-dissipation anomaly in the limit of vanishing viscosity. Various leading-order structural properties such as a novel $u_0$ dependence of a bulk length scale associated with front geometry are predicted in this limit. The analysis involves a formal analogy to random advection of diffusive scalars.

Appendices

Appendix A: Batchelor-Field Representation of Advected Huygens Fronts

The starting point for our analysis, as for CY, is the governing equation (9) for the scalar field h in three-dimensional space—a forced passive scalar with a nonlinear dissipation term representing Huygens propagation. Like CY, we infer that for single-scale flow, h has the properties of a Batchelor field [4, 7]—a regime of passive-scalar behavior that is more typically analyzed for scalars with linear (diffusive) dissipation. We consider Batchelor behavior of the h field to be plausible because either kind of dissipation term has a similar local effect of damping out scalar fluctuations below a small-scale cutoff ($r_0$), and the key cascade structure arises in the “viscous-convective” scale range $L> r > r_0$ where the effect of scalar dissipation is negligible. In any case, we adopt this viewpoint to demonstrate that passive-scalar analysis of the kind invoked by CY is consistent with the expected advection-propagation behavior once the relevant statistical properties of the Batchelor field are properly identified, modifying the CY results.

Despite the well-known Batchelor spatial correlation function proportional to $\ln (L/r)$, no direct connection appears to have been recognized in the literature between passive scalars in fluid dynamics and the recent mathematical studies of log-correlated fields. However, turbulent fields with multifractal intermittency have been analyzed as exponentials of log-correlated fields [2, 11].

Fig. 2

Representation of Batchelor scalar cascade in terms of order-unity scale bands

The key physical picture underlying the Batchelor field is advective transport and scale reduction by successive independent size-L eddies on the time scale $\tau$. This results in the following interlinked behaviors:

• Forcing (by transport along the unit mean gradient) injects scalar fluctuations of amplitude L at the spatial scale L.

• Advective strain causes stretching and folding of the scalar isosurfaces over time $\tau$, transferring existing fluctuations to spatial scales that are smaller by an order-unity factor (scalar cascade).

• Markers on the stretching isosurfaces separate over time $\tau$, increasing the number of surface elements that are subsequently dispersed independently by advective transport.

• Below the scale $r_0$, scalar fluctuations are dissipated within time $\tau$, terminating the cascade.

• The accumulated scalar fluctuations over the cascade equilibration time $T = \tau \ln (L/r_0)$ result from a large number $m = \ln (L/r_0)$ of independent eddy effects, and thus approximate a Gaussian random field by the central limit theorem [7].

A spectral diagram of h according to the Batchelor picture is shown in Fig. 2. This diagram is independent of the spatial dimension in which h is defined, and corresponds to spatial “pink noise” [11] (the spectral view of log-correlation). When restricted to two dimensions, the Batchelor field can be treated as a 2DGFF.

The fields G and h are equivalent with respect to the noted cascade behaviors but have different large-scale boundary conditions. Whereas G labels fronts in terms of their initial streamwise location, the value of $h = G + z - u_T t$ measures the streamwise displacement of each fluid element relative to a reference plane that moves with its front—rendering the statistics of h homogeneous and quasi-steady. Based on the equilibration time of the front, this means that h also estimates the net streamwise advection of the fluid element over the preceding time interval T (displacements older than this have been dissipated).

The extreme-value analysis of the h field that confirms ballistic scaling of the flame-brush width W can be viewed from several equivalent perspectives.

• 1. The argument based on material-surface advection in Sect. 2 characterizes W as the streamwise range occupied by an initially size-L surface element after advection for a time interval T. But the minimum and maximum of h over a size-L transverse surface element (i.e., the extrema of a 2DGFF) indicate the range of streamwise locations that have been advected to that surface element over the preceding interval T, which is statistically equivalent to W (time-reversed).

• 2. The direct correspondence of the structure of the 2DGFF h to the BRW description of material-surface advection can be seen as follows. Decompose h into its scale bands from Fig. 2: $h = h_1 + \cdots + h_m$, where $m = \ln (L/r_0)$ and each independent random field $h_i$ includes only inverse wavelengths between $e^{i-1}/L$ and $e^i/L$. In fact, we can relax the assumption that each $h_i$ is a Gaussian field. Then $h_1$ records streamwise advection by the most recent eddy (injection at scale L), and each successive $h_i$ records older streamwise displacements (age $i\tau$) that have been compressed to smaller scales by the intervening eddies. In a size-L surface element, $h_1$ contains a few undulations of amplitude L (typically ranging between $-L$ and $+L$) and thus represents the first branching of the (backward-in-time) BRW. Each successive $h_i$ contains a greater number of undulations of the same amplitude L on finer scales, representing continued branching. (For this reason, the effective number of independent degrees of freedom of the h field is exponential in m, but has no precise definition.) Just as the leading walker of the BRW can be estimated (bounded below) by greedy maximization that discards all but the current leading walker at each step, the maximum of the continuum h field can be estimated by “narrowing down” the relevant region as finer scales are incorporated. Within the size-L region, a size-L / e region can typically be found in which $h_1$ averages $+L$ (around the peak of an undulation). Within this size-L / e region, in turn, $h_2$ contains a few undulations and a size-$L/e^2$ region can typically be found in which $h_2$ averages $+L$. Ultimately, a size-$r_0$ region is found in which every $h_i$ equals roughly $+L$ and so h equals roughly $mL = L\ln (L/r_0)$. This confirms that the BRW result applies as a lower bound to leading order, even if h has non-Gaussian tails from incomplete convergence of the central limit theorem.

• 3. The argument in Sect. 3 uses the rigorous result [5] for extrema of a 2DGFF on a lattice (whose proof likewise leverages a BRW correspondence). We expect that identifying the lattice spacing with $r_0$ gives a result equivalent to the continuum h field with fluctuations damped below $r_0$. Indeed, the reasoning we have described here confirms this equivalence heuristically. Moreover, Sect. 3 provides another view of the relation of h to W, based on equating the range of h with the range of G over a size-L surface element (which is accurate to within $L \ll W$). Because G directly labels the fronts and has a unit mean gradient, the range of G is interpreted as the maximum streamwise offset that can exist between two flame brushes that both touch the surface element, which is thus at the trailing edge of one and the leading edge of the other. This makes clear that the range directly measures W.

Appendix B: Relationship Between the Present Analysis and the CY Methodology

As in Sect. 4, the CY analysis of their regime $\mathrm{A}'$ combines the relation $u_T = u_0 \langle |\varvec{\nabla } h| \rangle$ with an estimate of $\langle |\varvec{\nabla } h| \rangle$, which CY denote as $\langle | \delta h_{r_0} | \rangle / r_0$. Based on (10), which CY generalize by multiplying by a time-scale ratio that is of order unity in regime $\mathrm{A}'$ and thus immaterial, CY estimate that $\langle | \delta h_{r_0} | \rangle$ scales as $L \sqrt{\ln (L/r_0)}$ and hence as $W_D$ in present notation.

This result ostensibly follows from (10), and although the reasoning is not explained, it appears to be based on the assumption that $\langle | \delta h_{r_0} | \rangle$ scales as the square root of the spatial autocorrelation evaluated at $r = r_0$. However, for $d=3$ the relation (14) obeyed by the Batchelor spectrum implies $\langle | \delta h_{r_0} | \rangle = L$, with no logarithmic factor. (As explained in Appendix A, the associated heuristic is that the additive contribution of one cascade step to the range of h is L.)

Nevertheless, we first proceed using the CY estimate in order to reproduce their reported results. On this basis,

$$\begin{aligned} u_T = (u_0 L / r_0) \sqrt{\ln (L/r_0)}. \end{aligned}$$

(16)

Rather than using the estimate (4) of $r_0$, CY deduce $r_0$ by postulating

$$\begin{aligned} \langle | \delta h_{r_0} | \rangle = u_T T, \end{aligned}$$

(17)

where for regime $\mathrm{A}'$, their expression for T is equivalent to the first expression in (5), namely $T = \tau \ln (L/r_0)$. In the CY version of (17), $\delta h_{r_0}$ has a superscript (n) denoting the 1 / n power of the nth moment, but in regime $\mathrm{A}'$ there is no n dependence so the superscript is omitted here. (n dependence arises only in a high-$\mathrm{Re}$ intermittency-dominated regime that is actually the main focus of the CY analysis.)

(16) is based on the estimate $u_T = u_0 \langle | \delta h_{r_0} | \rangle / r_0$, indicating that $\langle | \delta h_{r_0} | \rangle / r_0$ cancels out of (17), which then gives $T = r_0 / u_0$ and thus $r_0 = u_0 \tau \ln (L/r_0)$. From this, CY deduce the leading-order dependence $r_0 = u_0 \tau \ln [L/ (\tau u_0)]$, corresponding to the expression for $r_0$ for regime $\mathrm{A}'$ in their Table I, which is $r_0 = u_0 \tau \ln (u_\mathrm{rms} / u_0)$ in present notation. The logarithmic factor in this expression for $r_0$ makes only a negligible high-order contribution when the expression is substituted into the argument of the logarithm in (16), so (16) and the expression for $r_0$ yield the result (1) to leading order. Specifically, this is the asymptotic result shown in Eq. 15 of CY, but in their Table I, their result for regime $\mathrm{A}'$ omits the square root sign, apparently inadvertently. (For completeness, we also note that n / L in the Table I result for regime C should instead read $\eta / L$.)

(17) is ostensibly a plug-flow relation which is therefore subject to the plug-flow consistency requirement that the flame-brush width on the left-hand side is no smaller than $W_D$. As noted above, the CY estimate of $\langle | \delta h_{r_0} | \rangle$ gives $W_D$ in present notation, which is marginally consistent, but the correct spectrally based result for $d=3$ is $\langle | \delta h_{r_0} | \rangle = L$, which is smaller than $W_D$ and therefore inconsistent. Instead, replacement of the left-hand side of (17) by $\langle | \delta h_L | \rangle$, which scales as the square root of the variance of h and thus as $W_D$, would be marginally consistent. In Sect. 2 it is shown that this choice recovers (1). CY obtained this result through the use of $\langle | \delta h_{r_0} | \rangle$ in (17) combined with its incorrect evaluation. Because the use of $\langle | \delta h_L | \rangle$ in (17) provides an error-free, physically intuitive rendering of the CY approach and results, it is adopted in Sect. 2 as a surrogate explanation of the approach.

The derivation in Sects. 2 and 3 yields a leading-order scaling of W that is unaffected by any hypothetical logarithmic factor in the parameter dependence of $r_0$. Such a factor would only contribute an additive sub-leading term to W. Thus, $u_T = u_\mathrm{rms}$ is recovered irrespective of any notional higher-order corrections to (4). Likewise, leading-order structural results in Appendix C are unaffected by such a correction.

As shown in Sect. 4 and above, $u_T$ can instead be directly evaluated from the relation $u_T = u_0 L / r_0$ that is based on estimation of the dissipation term, and then $u_T$ and T determine W. However, $u_T$ now depends to leading order on any hypothetical logarithmic factor subsumed in $r_0$. The derivation in Sects. 2 and 3 is thus more robust in this regard than the derivation in Sect. 4. The two derivations of $u_T$ agree if $r_0 = Lu_0/u_\mathrm{rms}$, hence supporting this scaling of $r_0$.

Appendix C: Streamwise Front Structure

To relate the BRW-based picture of front geometry in Sect. 5.1 to the G and h fields, we first note some exact properties of advection-propagation that follow from the G equation (7) and the existence of the quasi-steady state. Consider the specific front defined by the isosurface $G(t, {\mathbf {r}}) = 0$. The “progress variable” $c(t, {\mathbf {r}})$ is a field defined as the fluid state with respect to this front (0 for reactant, 1 for product): $c = \theta (G)$, where $\theta$ is the unit step function. By the chain rule, $\dot{c} = \delta (G) \dot{G}$ and $\varvec{\nabla } c = \delta (G) \varvec{\nabla } G$. Upon multiplying (7) by $\delta (G)$, we find

$$\begin{aligned} \dot{c} + {\mathbf {v}} \cdot \varvec{\nabla } c = u_0 |\varvec{\nabla } c|. \end{aligned}$$

(18)

This is of the same form as (7) due to the relabeling invariance of the G equation [21]. Moreover, we identify $|\varvec{\nabla } c| = \delta (G) |\varvec{\nabla } G|$ as the local surface density $\sigma$ of the front $G = 0$.

In the quasi-steady state, the ensemble average $\langle c\rangle$ (over flow realizations) is a function of $z - u_T t$, so averaging (18) gives

$$\begin{aligned} -u_T \nabla _z \langle c\rangle + \langle {\mathbf {v}} \cdot \varvec{\nabla } c\rangle = u_0 \langle \sigma \rangle . \end{aligned}$$

(19)

Also, we have

$$\begin{aligned} \nabla _z \langle c(t, {\mathbf {r}})\rangle = \nabla _z \bigl \langle \theta \bigl (G(t, {\mathbf {r}})\bigr )\bigr \rangle = \nabla _z \langle \theta (h - z + u_T t)\rangle = -\langle \delta (h - z + u_T t)\rangle , \end{aligned}$$

(20)

where the space-time arguments can be omitted from h because its single-point statistics are the same everywhere. Thus we obtain the exact result

$$\begin{aligned} u_T \langle \delta (h - z + u_T t)\rangle + \langle {\mathbf {v}} \cdot \varvec{\nabla } c\rangle = u_0 \langle \sigma \rangle . \end{aligned}$$

(21)

Now, we apply modeling assumptions to (21). As explained in Sect. 3, h is treated as a Batchelor field and therefore as approximately Gaussian. If we suppose that h is a Gaussian field, with variance $\langle h^2 \rangle = W_D^2$ from (13), then the corresponding Gaussian pdf gives

$$\begin{aligned} -u_T \nabla _z \langle c\rangle = u_T \langle \delta (h - z + u_T t)\rangle = \frac{u_T}{W_D} \exp \frac{-(z - u_T t)^2}{W_D^2}, \end{aligned}$$

(22)

and $\langle c\rangle$ itself has an error-function profile given by the cdf of h. Next, the advection term in (21) can be estimated via eddy diffusion (using $D = u_\mathrm{rms} L = u_T L$):

$$\begin{aligned} \langle {\mathbf {v}} \cdot \varvec{\nabla } c\rangle = -D \nabla ^2 \langle c\rangle = -u_T L \nabla _z^2 \langle c\rangle = -\frac{u_T L(z - u_T t)}{W_D^3} \exp \frac{-(z - u_T t)^2}{W_D^2}. \end{aligned}$$

(23)

Within the flame brush, where $|z - u_T t| < W = W_D^2 / L$, the term (23) is dominated by (22). Thus, from (21), the ensemble average surface density has a Gaussian profile,

$$\begin{aligned} \langle \sigma \rangle = \frac{u_T}{u_0 W_D} \exp \frac{-(z - u_T t)^2}{W_D^2}. \end{aligned}$$

(24)

The unimportance of the turbulent diffusion term (23) is consistent with plug flow. This derivation constitutes a more refined plug-flow picture that accommodates the Gaussian profile by accounting for dissipation throughout the front; by contrast, the basic plug-flow picture assumes that dissipation occurs only at the trailing edge, but the resulting surface profile is misleading unless it is explicitly filtered with an eddy-diffusion Gaussian, as described in Sect. 5.1. These pictures reflect different ways of understanding the quasi-steady state of advection-propagation, a complex nonlinear process involving a balance of surface growth, transport, and destruction.

Moreover, the present use of (21) in fact provides additional evidence for modeling h as a Gaussian field, at least within the noted range $|h| < W$. We observe that by definition,

$$\begin{aligned} \langle \sigma \rangle = \langle \delta (G) |\varvec{\nabla } G|\rangle = \langle \delta (h - z + u_T t)\rangle \biggl \langle \sqrt{|\varvec{\nabla } h|^2 - 2\nabla _z h + 1} \bigm | h = z - u_T t\biggr \rangle . \end{aligned}$$

(25)

Here we have reexpressed $\langle \sigma \rangle$ as the pdf of h times the conditional expectation of $|\varvec{\nabla } G|$, which is a valid relation regardless of whether h is Gaussian. (Because in the regime of interest $|\varvec{\nabla } h| \gg 1$, we could also approximate $|\varvec{\nabla } G| = |\varvec{\nabla } h|$.) However, under the assumption that h is Gaussian, the relation (25) allows us to conclude, as a purely geometrical property of Gaussian fields, that $\langle \sigma \rangle$ has a Gaussian profile. This follows from the homogeneous Gaussian-field property that, at a given point, $\varvec{\nabla }h$ and h are statistically independent, and so the conditional expectation in (25) is a constant equal to the unconditional expectation $\langle |\varvec{\nabla } G|\rangle = u_T/u_0$. Thus we reproduce the conclusion (24) without needing to invoke the dynamics of the G equation or (21). A similar geometrical result about the isosurface density of Gaussian fields has been obtained previously [28].

To see that this is a nontrivial self-consistency check on the Gaussianity of h, we note that constancy of the conditional expectation in (25) does not hold for generic (even homogeneous isotropic) random fields h. For example, if a candidate h field is constructed as a pointwise nonlinear function $h(\phi )$ of a Gaussian field $\phi$, then $\varvec{\nabla } h = (dh/d\phi ) \varvec{\nabla } \phi$, and so the conditional statistics of $\varvec{\nabla } h$ are field-dependent. This creates a contradiction between (25), which predicts that the $\langle \sigma \rangle$ profile deviates systematically from the pdf of h via dependence on this $dh/d\phi$ factor, and (21), which predicts that $\langle \sigma \rangle$ is simply proportional to the pdf of h (in the range where the advection term is negligible). However, while some non-Gaussian fields are thus excluded, this argument does not establish that a Gaussian h field is the only self-consistent possibility.

We emphasize that in fact h cannot be exactly Gaussian in all its statistics (see Sect. 3.2). Rather, the assertion is that the statistics of h being addressed (conditional statistics of gradients, and pdf excluding the very far tail) are likely asymptotically Gaussian to the precision captured in the reasoning presented—e.g., up to the neglect of the advection term in (21). The expected non-Gaussian features are those involving small relative modifications to h and $\varvec{\nabla } h$ or involving very rare events (rarer than those relevant to the extreme-value analysis). We have noted in Sects. 3.2, 4, and Appendix A that ballistic $u_T$ scaling is obtained from estimating $\langle |\varvec{\nabla } h|\rangle$ and from extreme-value analysis, even allowing for these plausible non-Gaussian features.