Reactive Rayleigh-Taylor Turbulence

Reactive Rayleigh-Taylor Turbulence

M. Chertkov, V. Lebedev, and N. Vladimirova

The Rayleigh-Taylor (RT) instability develops and leads to turbulence when a heavy fluid falls under the action of gravity through a light one. We consider this phenomenon accompanied by a reactive transformation between the fluids, and study with Direct Numerical Simulations (DNS) how the reaction (flame) affects the turbulent mixing in the Boussinesq approximation. We discuss “slow” reactions where the characteristic reaction time exceeds the temporal scale of the RT instability, . In the early turbulent stage, , effects of the flame are distributed over a maturing mixing zone, whose development is weakly influenced by the reaction. At , the fully mixed zone transforms into a conglomerate of pure-fluid patches of sizes proportional to the mixing zone width. In this “stirred flame” regime, temperature fluctuations are consumed by reactions in the regions separating the pure-fluid patches. This DNS-based qualitative description is followed by a phenomenology suggesting that thin turbulent flame is of a single-fractal character, and thus distribution of the temperature field is strongly intermittent.

1 Introduction

The Rayleigh-Taylor (RT) instability Rayleigh (1883); Taylor (1950); Chandrasekhar (1961) occurs in many natural and man-made flows. Under constant acceleration, produced e.g. by gravity, the instability develops into the RT turbulence Duff et al. (1962); Sharp (1984), characterized by a mixing zone which continuously grows. Many recent experimental Dalziel et al. (1999); Wilson & Andrews (2002), numerical Young et al. (2001); Dalziel et al. (1999); Cook & Dimotakis (2001); Ristorcelli & Clark (2004); Cabot & Cook (2006); Vladimirova & Chertkov (2008) and theoretical efforts Chertkov (2003); Chertkov et al. (2005) — later ones in the spirit of the turbulence phenomenology Kolmogorov (1941); Obukhov (1949); Corrsin (1951); Frisch (1995) — have been devoted to analysis of turbulence inside the mixing zone. The mixing fluids can also be involved in chemical or nuclear reactions, e.g. flame. A flame front propagating against the acceleration is modified by the RT instability, which leads to a reactive turbulence in the mixing zone. Studies of the externally stirred turbulent flames belong to the general field of turbulent combustion Damköler (1940); Williams (1985); Peters (2000); Poinsot & Veynante (2005); Borghi (1988); Kerstein (2001) and have many important applications. The buoyancy driven, reactive turbulence is believed to be the dominant mechanism for thermonuclear burning in type-Ia supernovae Khokhlov (1995); Gamezo et al. (2003); Zingale et al. (2005). The interplay of buoyancy, reactive transformation and turbulence also plays a crucial role in studies of fusion Freeman et al. (1977) and large-scale combustion, such as furnaces and fires Cetegen & Kasper (1996); Tieszen (2001). Turbulence, generated by buoyancy, facilitates mixing, and thus counteracts the dynamical separation of fluids (phases) due to reaction. To explain this competition between separation and mixing is the main challenge emerging in the description of reactive flows Kerstein (2001), which also applies in the broader context of multi-phase fluid mechanics, e.g. the RT turbulence of an immiscible mixture controlled by surface tension Chertkov et al. (2005).

To clarify the nature of the competition between separation and mixing, let us consider a typical reactive RT setting. Initially, the heavy, cold reactant is placed on top of the light, hot product. As in the non-reacting RT instability, the mixing zone develops. Since locally the reaction occurs only in the mixed fluid, turbulence inside the mixing zone enhances the cumulative reaction rate. The effect of reaction, however, is the opposite – the reaction consumes mixed material, and potentially limits the growth of the mixing zone. While diffusion and turbulence mix the reactant with the product, the reaction converts the mixed fluid into the product, thus causing an upward shift of the mixing zone as a whole.

Existing studies of reactive RT setting have mainly focused on the initial stages of instability Lima et al. (2006), or on the instability restricted by container walls in experiments Bockmann & Muller (2004) and by domain sizes in simulations Khokhlov (1995); Zingale et al. (2005); Vladimirova & Rosner (2005). The transverse (to direction of gravity) confinement results in a constant (on average) cumulative reaction speed, presumably dependent on the domain size Khokhlov (1995). In this system fluctuations, e.g. in velocity and density fields, do not grow with time. It is still not clear whether reaction can stabilize the unconfined RT instability. One suspects that the answer might depend on the relation between the timescale of the RT instability and the reaction timescale.

In this paper, we study RT turbulence unconfined by walls in the presence of reactions slower than RT instability. Armed with DNS and building a unifying phenomenological description, we analyze the interplay of turbulent mixing and reaction-mediated separation. Specifically, we focus on predicting macroscopic and microscopic features of the mixing zone, such as the upward transport of the mixing zone as a whole and statistics of density fluctuations within the mixing zone.

We examine the interacting effects of buoyancy and reaction in the simplest, however physically relevant setting, the Boussinesq approximation, where the variations in the fluid density are small and are related linearly to the temperature contrast. We assume that the pressure increase due to reaction is negligible, and that the Mach number is small and, consequently, the fluid is incompressible. Thus, our system can be modeled by the incompressible Navier-Stokes equation coupled to the advection-reaction-diffusion equation governing the temperature evolution. The set of equations is written as


where is the flow velocity, is the pressure, and represents temperature variations and/or the chemical composition of the fluid. It is convenient to choose equal to zero in the cold (heavy) phase and to unity in the hot (light) phase. Often, is referred to as the reaction progress variable. The reaction and production of heat is represented by the first term on the right hand side of Eq. (1); we refer to as the reaction rate function, and to as the characteristic reaction time, or reaction time scale. We consider a single step reaction (see Poinsot & Veynante 2005), where the reaction rate is zero in the pure phases, i.e. and for . The buoyancy force, i.e. the last term on the right hand side of Eq. (1), is directed along the gravitational acceleration . The factor is the Atwood number, where and are densities of the reactant and the product, respectively. We assume that the kinematic viscosity in Eq. (1) and the diffusion coefficient in Eq. (1) are comparable, and thus the Prandtl number is of the order of unity.

We compare our simulations with the benchmark non-reacting case, corresponding to in Eq. (1). Phenomenological theory introduced by Chertkov et al. (2003; 2005) and hereafter called the “phenomenology” suggests that in the non-reacting RT turbulence, velocity and density fluctuations are driven by buoyancy at the scales and , respectively, both comparable to the mixing zone width, . These fluctuations generate cascades (of energy and temperature) towards small scales. The cascades terminate at the viscous scale, , and (thermal) diffusive scale, , respectively Chertkov (2003); Chertkov et al. (2005). The scales and are of the same order provided the Prandtl number is of order unity. Previously conducted simulations of the non-reactive RT turbulence in the Boussinesq regime Vladimirova & Chertkov (2008) show that while the mixing zone width grows in time, and decrease with time. It was also found that the correlation radii of both the velocity and temperature fluctuations, and grow in time as . One special focus was on resolving the internal structure of the mixing zone more systematically than in the previous studies. It was shown that spatial correlations do not depend much on the vertical position within the mixing zone. In particular, this means that the relevant values of the major scales characterizing a snapshot of the RT turbulence depend only weakly on the height of the horizontal slice within the mixing zone.

2 Preliminary Considerations

In this paper we discuss the regime of relatively slow reaction: the time scale of the non-reacting RT instability, , and the characteristic reaction time, , are assumed to be well separated, .

At the early stage, , development of the mixing zone and the range of turbulent scales inside the mixing zone are weakly influenced by the reaction. Nevertheless a cumulative effect of the reaction can be seen in the overall shift of the mixing zone, as a whole, from the product side to the reactant side. To estimate the shift of the mixing zone, , let us integrate Eq. (1) along . For the first term on the left hand side one arrives at , while the only other nonzero contribution is associated with the reaction term on the right hand side. Since in this regime the heat production is determined mainly by the mixing on the scale of the whole mixing layer, the reactive contribution is estimated simply as . Thus, . Even though the overall shift of the mixing zone at is smaller than , grows faster than , so that the two scales become comparable at . From this time onward the reaction effects become more prominent.

The reaction becomes more effective when the turbulent turnover time at , estimated as , exceeds the characteristic reaction time . In this regime the reactant entrained into the mixing zone is burned out completely, and the overall shift of the mixing zone, , should be of order . This transition in the mixing zone evolution is also accompanied by a qualitative modification of the temperature fluctuations in the interior of the mixing zone. In contrast, the velocity distribution at all the scales is expected to show only weak sensitivity to the reaction even at .

The two stages observed for and will be called the mixed stage and the segregated stage, respectively. In the field of turbulent combustion (see Williams 1985; Peters 2000), the mixed regime is called “well-stirred reactor” Poinsot & Veynante (2005), “thickened flame” Borghi (1988) or “broken reaction zone” Peters (2000) while the segregated regime is the regime of “distributed reaction zones” Poinsot & Veynante (2005), “wrinkled-thickened flame” Borghi (1988) or “thin reaction zone” Peters (2000). Transition between the regimes occurring at can be interpreted as crossing the line on the turbulent combustion diagram (see Poinsot & Veynante 2005). Indeed, the Damköler number Damköler (1940), , is defined as the ratio of the integral time scale to the reaction time scale, , which grows linearly in time for the RT turbulence. The other popular dimensionless number in turbulent combustion, the turbulent Karlovitz number, , describes transition between the regime of “distributed reaction zones”/“wrinkled-thickened flame”/“thin reaction zone” and the “flamelet” regime. The Karlovitz number is defined as the ratio of the chemical time scale to the Kolmogorov time scale. For the slowly reacting RT turbulence studied here, the “flamelet” regime does not apply, i.e. is always much larger than unity and it increases with time. The laminar flame regime is also not observed in the systems considered here; consequently the laminar flame thickness is an irrelevant scale.

Figure 1: Schematic representation of the system evolution in log-log coordinates: the width of the RT mixing layer , the large scale fluctuations of velocity and temperature and , the shift of the mixing layer due to reaction, the turbulent flame thickness , and the dissipation scales and . The stages of the early RT instability development and transition to turbulence are shown shaded. is the characteristic reaction time, separating well-mixed and segregated regimes. In the mixed regime, the turbulent flame thickness corresponds to the typical width of interfaces, separating pure phase domains of size . The temporal scalings are phenomenological propositions discussed in Sections 24.

At the main effect of the reaction on the mixing zone is in creation of domains of pure phases ( and ). In these domains the fluctuations in the reaction term of Eq. (1) dominate fluctuations in the respective advection term, leading to complete suppression (burning out) of temperature fluctuations. The domains are separated by relatively thin (compared to domain size) interfaces where burning occurs. The interface width, also referred to as the turbulent flame thickness, , (that is the scale where the turnover time of velocity fluctuations is comparable to ) becomes much smaller than at due to the dominance of reaction.

Formation of the conglomerate of pure reactant and product domains should result in a qualitative transformation of the single-point statistics of the temperature field inside the mixing zone. A single peak distribution centered around transforms into a distribution with two peaks, related to the emergence of pure phase domains corresponding to and . The qualitative differences in the projected stages are summarized in Fig. 1 where the temporal behavior of different characteristics is shown schematically.

3 Numerical Results

Figure 2: Vertical and horizontal (, center of the mixing zone) slices of temperature for three different reaction strength.

The computational technique used in this work is similar to that described in Vladimirova & Chertkov (2008). In our simulations, we use the popular Kolmogorov-Petrovskii-Piskunov (KPP) model Fisher (1937); Kolmogorov et al. (1937), where with . The KPP model is not the only possible choice of the function . However, compared to other models, the KPP reaction provides a relatively thick laminar front which can be simulated at coarser resolution and makes computations more affordable. As before, we restrict ourself to the case of , and chose the units where and . The laminar flame parameters for five simulation sets discussed below are shown in Table 1.

A 1600.00 160.00 0.10

177.78 53.33 0.30

64.00 32.00 0.50

40.32 25.40 0.63

16.00 16.00 1.00

Table 1: Laminar flame parameter for simulations discussed in the text: reaction time , laminar flame thickness , and laminar flame speed, , for KPP reaction.

We solved the equations (1,1) using the spectral element code developed by Tufo & Fischer (2001). We use elements of size with 12 collocation points in each direction. The size of our computational domain is physical units, or collocation points. We stop our simulation at time , when the width of the mixing layer approaches the size of the computational domain. The boundary conditions are periodic in the horizontal () directions and no-slip in vertical () direction. The initial conditions, taken at , include a quiescent velocity and a slightly perturbed interface between the domains, determined by . The function describes the density profile across the interface and is a perturbation having the spectrum with modes with spectral index 0. Here the spectral index refers to the exponent of the wave-number (see Ramaprabhu et al. 2005), and it describes the shapes of the perturbation spectra.

The least processed results of numerical simulations, snapshots of the temperature field , are shown in Fig. 2; they are quite informative for our intuition. In case A () the structure of the mixing layer is practically indistinguishable from the non-reacting case (see Vladimirova & Chertkov 2008), at least up to . In case C () the reaction starts to influence the temperature distribution somewhere between and , where we observe emergence of regions of pure phases. In case E () reactants and products are well separated by relatively thin interfaces at . Thus, a transition from the mixed stage to the segregated stage occurs at , in accordance with the preliminary discussion of Section 2.

In Fig. 2 we observe that the width of the mixing layer, defined as the vertical distance between bubbles and spikes, is approximately the same in all cases. Measurements of the half-width of the mixing layer, defined as (where the overbar indicates averaging in a horizontal plane), confirm that is weakly dependent on the reaction rate (see Fig. 3a). In Fig. 3b we plot the root mean square (RMS) of the vertical velocity and see that in all five cases the RMS velocity is practically the same. Moreover, all five cases exhibit similar internal structure of the velocity field inside the mixing layer, as we see from the comparison of the temperature and velocity correlation lengths, and , measured in the central slice of the mixing zone (Fig. 3c). Following our earlier approach Vladimirova & Chertkov (2008) we define the velocity correlation length, , as the half-width of the the normalized two-point pair correlation function of velocity, , with . The temperature correlation function is defined in a similar way, but based on the temperature fluctuations, . Both temperature and velocity correlation lengths differ by 20-30% compared to each other for ranging from to . Similarly to what was reported in Vladimirova & Chertkov (2008) for the non-reacting case, . The viscous length decreases slowly with time in all cases, with less than variation between and . The main conclusion here is that the velocity field is essentially insensitive to the reaction in the slow reaction regime.

Figure 3: (a) half-width of mixing layer ; (b) RMS of vertical velocity at ; (c) pumping scale (dashed lines) and scale (solid lines) measured at ; (d) bulk burning rate (solid lines) compared to growth rate of RT instability (dashed dots); (e) burning efficiency measured as ratio (solid lines), the average reaction rate in the middle slice (dashed lines), and well-mixed fraction of the slice area, (dotted lines); (f) turbulent flame thickness in the middle slice obtained using “expanding circle” technique. Arrows in (d)-(f) show the reaction time.

Next we look at the shift of the mixing layer , introduced in Section 2. We compute the cumulative reaction, or bulk burning rate, , and compare it to the growth rate of the mixing layer, . Both quantities are shown as functions of time in Fig. 3d. We see that the bulk burning rate is greater for higher reaction rates. The time when and become comparable is approximately equal to .

It is not surprising that the bulk burning rate increases with decrease in . However, in addition to this obvious effect, we need to take into account the reaction efficiency, which depends on how well the fluids are mixed within the mixing layer. In Fig. 3e we show reaction efficiency of the whole mixing layer, , and compare it to the reaction efficiency in the middle slice, . Good agreement observed between the two characteristics suggests that the fraction of well-mixed fluid within the mixing zone does not vary significantly with the vertical position. In fact, we had also made this observation earlier in our study of the non-reacting case Vladimirova & Chertkov (2008), where the mixing efficiency stayed at the constant level of across the whole layer. In addition, Fig. 3e shows the fraction of well-mixed fluid, , defined as a relative area of the slice occupied by fluid with . We see that behaves similar to ; some differences reflect an arbitrary choice of the selected limits. The area fraction , as well as the average reaction rate in the middle slice and the reaction efficiency characterize the transition from the mixed stage to the separated stage. In the mixed stage is of order unity, and in the segregated stage it is a decreasing function of time. Numerical results shown in Fig. (3e) confirm a decrease of observed for .

Since the reaction at the central, , slice describes the bulk burning rate so well, we pay particular attention to the temperature distribution within the slice, shown in Fig. 4. In case A the probability distribution function (PDF) of temperature evolves in time almost as if there were no reaction at all. It starts from a single peak distribution determined by the initial conditions (slightly perturbed diffused interface). Then, during the linear stage of the RT instability, , the PDF transforms into the two-peak shape and becomes single-peaked again at . The transformation from one to two peaks corresponds to the transition to the non-linear regime of the RT instability, associated with secondary Kelvin-Helmholtz type shear instability and formation of RT mushrooms, while transition from two peaks to one corresponds to the destruction of RT mushrooms and formation of the turbulent mixed zone. When reaction is stronger, as in cases B and C, we observe yet another transformation in the PDF, namely the appearance of a narrow peak around and decrease in the middle of the range at . The transition to the two-peaked shape at is of interest to us since it indicates transition to the segregated regime. (In cases D and E, this 1-2-1-2 peak pattern is not noticeable because is of the order of, or shorter than, the transition time from linear to non-linear RTI.) Quantitatively the process of the transition to the segregated regime can be described by integrating the PDF function over some interval in the vicinity of . Selecting the interval , we obtain , described earlier.

Figure 4: PDF of temperature in the middle slice. Panels (a)-(e) show time evolution of PDFs. Panel (f) compares PDFs at for different reaction times, .

The well-mixed fraction, , is related to the turbulent flame width, , or the width of the layer separating the pure-phase domains. Aiming to study the geometry of the turbulent flame we measure in the following way: for each point with we find the largest radius of the circle around this point containing only points with ; this length, averaged and multiplied by four, represents the typical width of the burning region, , shown in Fig. 3f. We observe that (i) increases with , (ii) for (compare Fig. 3f and Fig. 3c), and (iii) for , grows with time but slower than , so decreases with time.

To summarize, we have observed in our numerics the transition at from mixed to segregated regimes in variety of ways, from visual comparison of temperature fields and temperature PDFs to direct measurements of reaction efficiency and turbulent flame thickness.

4 Phenomenology

This Section focuses on analysis of the asymptotic (), segregated regime, supposedly characterized by extended inertial interval and scaling behaviors. The simulations discussed above do not run long enough to reach and accurately resolve this regime. However, the simulations are in a qualitative agreement with the phenomenology constructed, thus providing a starting point for more ambitious simulations in the future.

We first review the relevant facts from non-reacting RT turbulence, (). In this case the phenomenology Chertkov (2003), supported by numerical simulations Cabot & Cook (2006); Zingale et al. (2005) and some (though limited) experimental observations, can be summarized as follows. At the width of the mixing zone grows as . Velocity fluctuations within the mixing zone are described by the Kolmogorov cascade, whereas temperature fluctuations follow the Corrsin-Obukhov cascade. Both cascades are driven by the largest RT scale (the spikes and bubbles) at . Even though this driving scale grows with time, turbulence at smaller scales is adiabatically adjusted to the growth. The adiabaticity is due to the monotonic decrease of the turbulence turnover time with the scale.

As discussed above and illustrated in Fig. 1, for the development in the reacting case proceeds as in the benchmark non-reacting case. Here is well mixed within the mixing zone, and the single point distribution of the temperature is peaked around the median value of . Typical fluctuations of the velocity and temperature inside the mixing zone (measured in terms of the differences and between two points separated by a distance ) are described by the Kolmogorov-Corrsin-Obukhov estimates


which are insensitive to the reaction. The viscous and thermal diffusion scales, and , can be estimated as , provided the Prandtl number is of order unity.

Since the velocity correlation functions are formed by the direct cascade determined by the non-linear term in Eq. (1), the driving effectively occurs at the scale , with the former one being not very sensitive to the reaction statistics, and thus determined by the same estimates as in the non-reacting case, Eq. (4). (Weak sensitivity of the velocity statistics to the reaction was also observed in our simulations, see Fig. 3.) In contrast, the statistics are strongly modified at when pure phase regions are formed separated by relatively thin interfaces (see Figs. 3 and 4). The interface width is estimated by balancing advection and reaction terms in Eq. (1), . One thus derives the following estimate for the turbulent flame thickness,


The turbulent flame thickness does grow with time, but in such a way that both and also grow as time advances. (Although our numerical data do not run long enough to illustrate the increase of with time, we did observe the increase of with . See Fig. 3f.)

Let us now exploit the Kolmogorov-Obukhov relation, calculating the time derivative of in accordance with Eq. (1), and then averaging the result over a slice perpendicular to the -axis for a fixed value of , where . Making use of the adiabaticity and neglecting the reactive and diffusive terms in Eqs. (1,1) one derives

where the flux term on the right-hand side originates from driving at the large-scale, , due to the large scale temperature gradient set by buoyancy Shraiman & Siggia (1994). Using the Kolmogorov estimation (4) for , one arrives at


The most important assumption, made while deriving Eq. (4.0), concerns neglecting the reactive contribution into the balance. This assumption will be justified later in the Section.

At , Eq. (4.0) also describes the probability for two points separated by to fall in two distinct domains of and , or vice versa. Exactly the same probabilistic arguments apply to a temperature structure function of any positive order, thus resulting in the asymptotic independence of the respective scaling exponent of the order:


The expression implies strong intermittency of the temperature field, as the ratios all grow with scale at . Notice that these arguments also apply to the immiscible RT, of the type discussed in Chertkov et al. (2005).

The expressions (4.04.0) suggest that the flame interface is fractal and, moreover, single-fractal (as opposed to multi-fractal, see e.g. discussion of Sreenivasan et al. (1989) and Kerstein (1991)). The fractalization of the flame is properly explained by dependence of the fraction, , of the mixing zone on the turbulence flame thickness, . Taking Eqs. (4.04.0) at , one estimates, . This also implies that , thus confirming the general conclusion made in Section 2.

Now we are ready to justify the approximation leading to Eq. (4.0). Let us estimate contribution of the flame/reaction term, , into the aforementioned balance relation. The product is non-zero only if falls inside the interface. Then is much less than since is of order unity and it also has zero average. The average is determined by the fraction of the well mixed region within the mixing zone, . One concludes, that, indeed, at , thus justifying the assumption made above in deriving Eq. (4.0) and subsequently Eq. (4.0).

Turning to discussion of the range of scales smaller than but larger than , one notices that the reaction term does not contribute to the balance, as it is much smaller than the respective advection contribution. At these scales, where the direct effect of the reaction term is irrelevant and the Kolmogorov velocity estimates still survive, one expects that the Obukhov-Corrsin scaling, , applies. Matching the self-similar behavior with Eq. (4.0) at , one derives the following scaling relation for the structure functions at


This formula allows a simple interpretation: the first term on the right hand side stands for the fraction of the well mixed region within the mixing zone, , where the reaction takes place, while the second term accounts for the Corrsin-Obukhov decay of correlations with the decrease in scale. The main contribution in the RHS of Eq. (4.0) comes from the interfacial domains, while contributions associated with pure phase domains, where or , is much smaller.

5 Conclusions

In this manuscript we analyze the reactive RT turbulence in the case of a slow flame realized when the typical reaction time is larger then the time of RT instability development, . The RT instability leads to formation of the mixing zone at . Further development of the mixing zone is roughly split into the early turbulent stage, , characterized by well-mixed hot and cold phases, and the later stage, , where pure state phases are segregated. In the early stage, the system is only mildly sensitive to the reaction (flame), resulting in a slight shift of the mixing zone as a whole, but no visible feedback of the flame on velocity distribution. The velocity fluctuations are still insensitive to the reaction, at , however the temperature fluctuations become modified significantly. In this regime, burning takes place within thin interfaces separating large patches of pure phases. Our numerical simulations confirm the basic prediction of the transition at . All the qualitative features expected from the transition are observed in the simulations. Given that our numerical resources were not sufficient for detailed analysis of the asymptotic, , stage, we rely here on the phenomenology. Separation of the advective range for density fluctuations in two ranges, and formation of a single-fractal flame interface are our two main phenomenological predictions. This highly intermittent behavior of density fluctuations is in a great contrast with what was observed without reaction. These and other predictions of the phenomenology are subjects for juxtaposition with future numerical and laboratory experiments.

Notice that the general picture of mixing at discussed above is reminiscent of the immiscible turbulence discussed in Chertkov et al. (2005). In both cases large regions of pure phases are separated by thin interfaces. Viewed at the scales larger than the interface width, the interfaces are advected passively. Moreover, strong intermittency in the density fluctuations, reported above for the reactive case, also takes place in the setting of externally stirred (e.g. by gravity) immiscible turbulence.

Finally, our phenomenology also applies to other regimes of turbulent flames, e.g. those realized in combustion engines, where the turbulent flame width is positioned in between the integral scales of turbulence and the viscous/diffusive scales, .

6 Acknowledgments

We wish to thank P. Fischer for the permission to use the Nekton code, A. Obabko and P. Fischer for the detailed help in using the code, and V.G. Weirs for useful comments. This work was supported by the U.S. Department of Energy at Los Alamos National Laboratory under Contract No. DE-AC52-06NA25396, under Grant No. B341495 to the Center for Astrophysical Thermonuclear Flashes at the University of Chicago, and RFBR grant 06-02-17408-a at the Landau Institute.

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