2.1. Earlier Transit Stage
Let us explore the motion of an infinitely thin front in statistically stationary, homogeneous, isotropic turbulence. The front self-propagates locally normal to itself at a constant speed
. The turbulence is (i) unaffected by the front, (ii) characterized by an asymptotically high turbulent Reynolds number
, and (iii) described by the Kolmogorov theory [
4,
5,
23]. The front is considered to propagate slowly, i.e., the Kolmogorov velocity
is much larger than
. In this section, we address the early transient stage, i.e.,
, of the evolution of an initially (
) planar front.
The present analysis is based on the theory of surface area growth, developed by Batchelor [
24] for an infinitesimal element of a material surface. The analysis is also based on the results of DNS studies [
25,
26] of the same phenomenon, and theoretical and DNS results [
27] on the growth of the area of a finite-length element of a material surface, i.e., an element whose area is much larger than
. We also use the theory of turbulent diffusion, developed by Taylor [
28]. As we will see later, the earlier transit stage takes a time interval much shorter than the eddy turnover time
. During such a short time interval, the area growth rates are almost the same for the infinitesimal and finite-length elements of a material surface [
27]. This is because the folding of the finite-length elements, caused by large-scale eddies, is a relatively slow process.
When a planar material surface—which is normal to the
-axis (streamwise direction in the following)—is embedded into the Kolmogorov turbulence, the surface adapts itself to the flow field during a short transient time interval of
. Subsequently, the mean (ensemble-averaged) surface area
is expected to grow exponentially with time [
24,
25,
26], i.e.,
where
is a constant at about 0.28 [
25,
26],
designates the Kolmogorov time scale, and
is the area of the considered element of the initial planar material surface at
, i.e.,
. Note that a subsequent DNS study [
27] indicates that, due to the folding of finite-length material surface elements during a later stage,
in a range of
when
.
At the same time, the streamwise dispersion
of a material surface grows linearly with time [
28].
at
, with a similar linear dependence of a mean turbulent flame brush thickness on flame-development time being documented in various experiments reviewed elsewhere [
29]. The constraints of
and
are consistent with one another in the considered case of
.
As argued by Yeung et al. [
26], Equation (1), which holds for an infinitesimal element of a material surface, describes also the evolution of the area
of an infinitesimal element of a dynamically passive front, provided that
and
Moreover, during the studied short earlier stage (
), the same equation holds for finite-length surface elements [
27], as already noted earlier. Thus, if
at
.
Furthermore, the already cited DNS study by Yeung et al. [
26] shows that, during the considered time interval (
), the distance between the material and the self-propagating surfaces that coincide at
is smaller than the Kolmogorov length scale
with a high probability. This feature is associated with the well-known fact that positive rates of strain of a material surface statistically dominate in the Kolmogorov turbulence. Indeed, if the local rate of strain of a material surface is positive, the locally normal (to the surface) velocity vector
points to the surface, and the magnitude
is increased with distance from the surface. Consequently, the velocity
can be significantly larger than
already at a short distance from the surface. Therefore, as
, self-propagation of the front plays a minor role locally, the vector
points to the front, and the local flow impedes further divergence of the front and material surface.
However, there are fundamental differences between the two surfaces. Indeed, first, there is no cusp formation at a material surface and, second, the neighboring/adjoining elements of a folded (folds are produced by strong advection) material surface never collide. Therefore, (i) the area of a material surface is well known to grow exponentially with time, and (ii) the distance
between the surface elements can be very low. For instance, DNS data show that the distance
is randomly distributed in a wide range of length scales, which can be seen in Figure 6 in Ref. [
26]. In the case of a self-propagating front, cusp formation and collisions of its elements result in the local surface annihilation if the local distance between the neighboring front elements is small enough. However, during the studied short earlier stage, both effects may be neglected, as discussed earlier.
Let us compare the fluid volume consumed by the front at instant
with the volume of the streamwise turbulent dispersion of the front, i.e., a volume bound by the leading and trailing edges of the front. The former volume can be estimated as follows:
where
is the volume rate of the fluid consumption at instant
.
If
, the second term in square brackets is negligible, and
i.e., the volume of the consumed fluid is controlled by the small-scale turbulence and grows exponentially with time.
By virtue of Equation (4), the volume of the streamwise dispersion of the front is equal to
and, consequently, is controlled by large-scale turbulent eddies. This volume grows linearly with time at
, contrary to the exponential growth of
. Therefore, in spite of
at
, because
, the exponentially growing volume
and the linearly growing volume
should become equal to one another at certain instant
. In other words, at instant
, the fluid consumed by the front fills the volume formed by the streamwise dispersion of the front.
To estimate this critical instant, let us invoke the following simple constraint:
Henceforth, numerical factors are skipped for simplicity. Equation (8) can be rewritten as follows:
Here,
is the mean thickness of a thin layer consumed by the front, or, in other words, the mean distance between initially coinciding elements of the front and material surface. The symbol
designates the mean distance between opposed elements of either the front or the material.
Equations (3) and (6) yield
Both the distance
and the microscale
are much less than the Kolmogorov length scale, i.e., they are inside the dissipation subrange of the turbulence spectrum. This estimate agrees with the DNS data by Yeung et al. [
26], thus supporting the present study. Note that the microscale
will also play an important role in an analysis of the statistically stationary state of the front evolution, discussed in the next subsection.
Substitution of Equations (6) and (7) into Equation (8), or substitution of Equations (1), (3), (6), and (7) into Equation (9) yields.
Taking logarithm of Equation (11), we arrive at
Under the considered conditions of
, term
. Therefore,
, the last term on the right-hand side of Equation (12) may be neglected when compared to the left-hand side. Consequently, the non-linear Equation (11) has the following approximate solution:
By virtue of Equation (13), the following necessary condition
should be satisfied in order for
, which is required for Equations (2) and (4) to be valid.
At instant
, the front area given by Equations (3) and (11) is equal to
The turbulent consumption velocity is equal to
The volume of the consumed fluid is equal to
see Equations (7) and (8). Finally, the mean consumption velocity averaged over
is equal to
Independence of the mean consumption velocity on the Kolmogorov scales does not mean that the Kolmogorov eddies are unimportant. On the contrary, it is the Kolmogorov eddies that create front surface within the framework of the above analysis. Nevertheless, the outcome, i.e., the mean
, is independent of the Kolmogorov scales. This apparent paradox is basically similar to the well-known independence of the mean dissipation rate on viscosity in the Kolmogorov turbulence at
, or independence of the mean rate of entrainment of ambient irrotational fluid into turbulent fluid on viscosity in shear flows [
30]. While both the dissipation and entrainment occur due to viscosity, the mean rates of the two processes are controlled by large-scale velocity fluctuations at
, whereas small-scale phenomena adjust themselves to these mean rates. As noted by Tsinober [
6], “
small scales do the ‘work’, but the amount of work is fixed by the large scales in such a way that the outcome is independent of viscosity”.
2.2. Statistically Stationary State
The method used in
Section 2.1 to analyze the early (
) transient stage of front propagation under conditions of
is based on the hypothesis that a material surface and a self-propagating front that coincide at
are very close to one another (i.e., the distance between them is smaller than the Kolmogorov length scale) during a short (
) time interval. This hypothesis allows us to model temporal growth of the front surface area by invoking results that are well known for material surfaces. However, this hypothesis does not hold at
when the front area reaches a statistically stationary state. In this limit, the growth of the front surface area due to turbulent straining is counterbalanced by a reduction of the front surface area due to joint actions of folding of finite-length front elements, caused by strong advection, and subsequent collisions of self-propagating fronts. As a result, neighboring front surface elements collide, and the front surface area is reduced.
Here, to examine the statistically stationary regime of slow front propagation, we will show that the smoothing of small-scale wrinkles occurs in the dissipation range of the turbulence spectrum (i.e., at length scales smaller than the Kolmogorov scale). Accordingly, we will consider the front surface to be a bifractal, i.e., two fractals with different dimensions, associated with the dissipation and inertial ranges. A similar scenario was explored by Sreenevasan et al. [
20] when discussing turbulent mixing for Schmidt numbers far greater than unity, as portrayed in Figures 2a and 6 in the cited paper. Recently, such ideas were developed for a flame of a finite thickness [
31,
32]. In the present communication, the bifractal concept is applied to an infinitely thin front. In particular, to explore the influence of turbulent eddies on the area of a slowly (
) propagating front, the area response to small-scale and large-scale turbulent eddies is modeled by invoking two different fractal submodels. More specifically, both large-scale and small-scale wrinkles of the front are considered to be fractals, but with different dimensions (
and
and different cut-off scales. Moreover, the outer cut-off scale for the small-scale fractal is considered to be equal to the inner cut-off scale for the large-scale fractal. These two equal cut-off scales are called a crossover length scale in the following. Thus, the focus of the following discussion is placed on the two fractal dimensions, the crossover length scale, as well as the inner
and outer
cut-off scales for small-scale and large-scale wrinkles of the front surface, respectively.
First, following a common supposition [
20,
21,
22], the large outer cut-off scale
is assumed to be proportional to a turbulent integral length scale
.
Second, the crossover length scale is associated with the boundary between inertial and dissipation ranges of the turbulence spectrum. Therefore, the crossover length scale is proportional to the Kolmogorov length scale
. Thus, the large-scale fractal covers the following range
of wrinkle scales
. It is worth noting that
is considered to be the inner cut-off scale not only in single-fractal models of non-reacting turbulent flows [
20] or a bifractal model of turbulent mixing at a large Schmidt number [
20], but also in certain single-fractal models of highly turbulent flames [
33]. Contrary to the latter models, the front is hypothesized to be another fractal even at smaller length scales
, rather than a smooth interface. The point is that, under the considered conditions of an infinitely thin and slowly propagating (i.e.,
) front, there is no physical mechanism that can smooth the front surface at scales larger than the Kolmogorov length scale.
Indeed, third, the sole physical mechanism of smoothing small-scale wrinkles on the surface of an infinitely thin front consists of kinematic restoration due to the self-propagation of the front [
21,
22]. This is the key difference between the present study and a recently developed bifractal model [
32] of a highly turbulent reaction wave that has a mixing zone of a finite thickness. For such waves, the inner cut-off scale is controlled by molecular mixing [
32]. For an infinitely thin front, the small inner cut-off scale
is identified as the Gibson scale corresponding to the front velocity
. Therefore, the scale
is found using the following constraint:
where
designates the velocity difference in two points separated by the distance
.
The same constraint is adopted in the classical single-fractal models of turbulent flames [
21,
22], which address the case of
and, accordingly, estimate the velocity difference following the Kolmogorov scaling for the inertial interval [
4,
5], i.e.,
. However, under conditions of
examined here, the scale
belongs to the viscous (dissipation) subrange of the turbulence spectrum. Therefore, the difference
should be estimated using the Taylor expansion [
4]. Consequently, by retaining the linear term in the expansion, we arrive at
Equations (19) and (20) yield
A comparison of Equations (10) and (21) shows that the inner cut-off scale
is equivalent to the microscale
introduced in
Section 2.1. Obviously, the scales
and
differ from the common Gibson length scale
[
21,
22], which characterizes interaction of the front with turbulent eddies from the inertial range.
Fourth, the area of a bifractal surface is evaluated as follows [
20,
32]:
where subscripts 1 and 2 refer to the large-scale interval of
and the small-scale interval of
, respectively. In terminology by Sreenivasan et al. [
20],
is the area measured with resolution
, and
is the true front surface area increased jointly by large-scale and small-scale wrinkles.
Substitution of Equation (22) into Equation (23) yields
The value of the fractal dimension
of the small-scale wrinkles can be found by noting that the scales
are inside the dissipation subrange. Accordingly, the small-scale wrinkles of the front surface fill the space between
and
, and, hence,
[
34], as proposed by E. Hawkes during discussion with the first author in Dubrovnik in April 2017. For the fractal dimension
of wrinkles whose scale is larger than
, the common value [
20,
21,
22] of
may be adopted.
Subsequently, Equations (22) and (24) read
Thus, the turbulent consumption velocity is equal to
Finally, it is worth noting the following point. If we consider the entire small-scale (
) fractal to be a broadened front propagating at an increased speed
then the Gibson length scale for this front is equal to
, which, in its turn, is equal to the crossover length scale or the inner cut-off scale for the large-scale (
) fractal. This example shows self-consistency of the present estimates of the two inner cut-off scales
and
, as they both are associated with Gibson scales obtained by comparing the front speed and velocity difference for the appropriate range of the turbulence spectrum. Moreover, the turbulent consumption velocity is again equal to
. Indeed,