A. J. Aspden

Type Ia Supernovae: Calculations of Turbulent Flames Using the Linear Eddy Model

The nature of carbon burning flames in Type Ia supernovae is explored as they interact with Kolmogorov turbulence. One-dimensional calculations using the Linear Eddy Model of Kerstein elucidate three regimes of turbulent burning. In the simplest case, large-scale turbulence folds and deforms thin laminar flamelets to produce a flame brush with a total burning rate given approximately by the speed of turbulent fluctuations on the integral scale, UL , This is the regime where the supernova explosion begins and where most of its pre-detonation burning occurs. As the density declines, turbulence starts to tear the individual flamelets, making broader structures that move faster. For a brief time, these turbulent flamelets are still narrow compared to their spacing and the concept of a flame brush moving with an overall speed of UL remains valid. However, the typical width of the individual flamelets, which is given by the condition that their turnover time equals their burning time, continues to increase as the density declines. Eventually, mixed regions almost as large as the integral scale itself are transiently formed. At that point, a transition to detonation can occur. The conditions for such a transition are explored numerically and it is estimated that the transition will occur for densities near 1 × 107 g cm–3, provided the turbulent speed on the integral scale exceeds about 20% sonic. An example calculation shows the details of a detonation actually developing.

HIGH-RESOLUTION SIMULATIONS OF CONVECTION PRECEDING IGNITION IN TYPE Ia SUPERNOVAE USING ADAPTIVE MESH REFINEMENT

We extend our previous three-dimensional, full-star simulations of the final hours of convection preceding ignition in Type Ia supernovae to higher resolution using the adaptive mesh refinement capability of our low Mach number code, MAESTRO. We report the statistics of the ignition of the first flame at an effective 4.34 km resolution and general flow field properties at an effective 2.17 km resolution. We find that off-center ignition is likely, with radius of 50 km most favored and a likely range of 40-75 km. This is consistent with our previous coarser (8.68 km resolution) simulations, implying that we have achieved sufficient resolution in our determination of likely ignition radii. The dynamics of the last few hot spots preceding ignition suggest that a multiple ignition scenario is not likely. With improved resolution, we can more clearly see the general flow pattern in the convective region, characterized by a strong outward plume with a lower speed recirculation. We show that the convective core is turbulent with a Kolmogorov spectrum and has a lower turbulent intensity and larger integral length scale than previously thought (on the order of 16 km s–1 and 200 km, respectively), and we discuss the potential consequences for the first flames.

DISTRIBUTED FLAMES IN TYPE Ia SUPERNOVAE

At a density near a few ×10 7 gc m −3 , the subsonic burning in a Type Ia supernova (SN) enters the distributed regime (high Karlovitz number). In this regime, turbulence disrupts the internal structure of the flame, and so the idea of laminar burning propagated by conduction is no longer valid. The nature of the burning in this distributed regime depends on the turbulent Damk¨ ohler number (DaT ), which steadily declines from much greater than one to less than one as the density decreases to a few ×10 6 gc m −3 . Classical scaling arguments predict that the turbulent flame speed sT , normalized by the turbulent intensity ˇ u, follows sT / ˇ u = Da 1/2 T for DaT 1. The flame in this regime is a single turbulently broadened structure that moves at a steady speed, and has a width larger than the integral scale of the turbulence. The scaling is predicted to break down at DaT ≈ 1, and the flame burns as a turbulently broadened effective unity Lewis number flame. This flame burns locally with speed sλ and width lλ, and we refer to this kind of flame as a λ-flame. The burning becomes a collection of λ-flames spread over a region approximately the size of the integral scale. While the total burning rate continues to have a well-defined average, sT ∼ˇ u, the burning is unsteady. We present a theoretical framework, supported by both one-dimensional and three-dimensional numerical simulations, for the burning in these two regimes. Our results indicate that the average value of sT can actually be roughly twice ˇ u for DaT 1, and that localized excursions to as much as 5 times ˇ u can occur. We also explore the properties of the individual flames, which could be sites for a transition to detonation when DaT ∼ 1. The λ-flame speed and width can be predicted based on the turbulence in the star (specifically the energy dissipation rate e ∗ ) and the turbulent nuclear burning timescale of the fuel τ T nuc .W e propose a practical method for measuring sλ and lλ based on the scaling relations and small-scale computationally inexpensive simulations. This suggests that a simple turbulent flame model can be easily constructed suitable for large-scale distributed SNe flames. These results will be useful both for characterizing the deflagration speed in larger full-star simulations, where the flame cannot be resolved, and for predicting when detonation occurs.

The effect of sudden source buoyancy flux increases on turbulent plumes

Building upon the recent experimentally verified modelling of turbulent plumes which are subject to decreases in their source strength (Scase et al., J. Fluid Mech., vol. 563, 2006b, p. 443), we consider the complementary case where the plumes source strength is increased. We consider the effect of increasing the source strength of an established plume and we also compare time-dependent plume model predictions for the behaviour of a starting plume to those of Turner (J. Fluid Mech., vol. 13, 1962, p. 356). Unlike the decreasing source strength problems considered previously, the relevant solution to the time-dependent plume equations is not a simple similarity solution. However, scaling laws are demonstrated which are shown to be applicable across a large number of orders of magnitude of source strength increase. It is shown that an established plume that is subjected to an increase in its source strength supports a self-similar ‘pulse’ structure propagating upwards. For a point source plume, in pure plume balance, subjected to an increase in the source buoyancy flux F0, the rise height of this pulse in terms of time t scales as t3/4 while the vertical extent of the pulse scales as t1/4. The volume of the pulse is shown to scale as t9/4. For plumes in pure plume balance that emanate from a distributed source it is shown that the same scaling laws apply far from the source, demonstrating an analogous convergence to pure plume balance as that which is well known in steady plumes. These scaling law predictions are compared to implicit large eddy simulations of the buoyancy increase problem and are shown to be in good agreement. We also compare the predictions of the time-dependent model to a starting plume in the limit where the source buoyancy flux is discontinuously increased from zero. The conventional model for a starting plume is well approximated by a rising turbulent, entraining, buoyant vortex ring which is fed from below by a ‘steady’ plume. However, the time-dependent plume equations have been defined for top-hat profiles assuming only horizontal entrainment. Therefore, this system cannot model either the internal dynamics of the starting plumes head or the extra entrainment of ambient fluid into the head due to the turbulent boundary of the vortex ring-like cap. We show that the lack of entrainment of ambient fluid through the head of the starting plume means that the time-dependent plume equations overestimate the rise height of a starting plume with time. However, by modifying the entrainment coefficient appropriately, we see that realistic predictions consistent with experiment can be attained.

Cosmological fluid mechanics with adaptively refined large eddy simulations

We investigate turbulence generated by cosmological structure formation by means of large eddy simulations using adaptive mesh refinement. In contrast to the widely used implicit large eddy simulations, which resolve a limited range of length scales and treat the effect of turbulent velocity fluctuations below the grid scale solely by numerical dissipation, we apply a subgrid-scale model for the numerically unresolved fraction of the turbulence energy. For simulations with adaptive mesh refinement, we utilize a new methodology that allows us to adjust the scale-dependent energy variables in such a way that the sum of resolved and unresolved energies is globally conserved. We test our approach in simulations of randomly forced turbulence, a gravitationally bound cloud in a wind, and the Santa Barbara cluster. To treat inhomogeneous turbulence, we introduce an adaptive Kalman filtering technique that separates turbulent velocity fluctuations on resolved length scales from the non-turbulent bulk flow. From the magnitude of the fluctuating component and the subgrid-scale turbulence energy, a total turbulent velocity dispersion of several 100 km/s is obtained for the Santa Barbara cluster, while the low-density gas outside the accretion shocks is nearly devoid of turbulence. The energy flux through the turbulent cascade and the dissipation rate predicted by the subgrid-scale model correspond to dynamical time scales around 5 Gyr, independent of numerical resolution.

Burning thermals in type Ia supernovae

We develop a one-dimensional theoretical model for thermals burning in Type Ia supernovae based on the entrainment assumption of Morton, Taylor, and Turner. Extensions of the standard model are required to account for the burning and for the expansion of the thermal due to changes in the background stratification found in the full star. The model is compared with high-resolution three-dimensional numerical simulations, both in a uniform environment and a full-star setting. The simulations in a uniform environment present compelling agreement with the predicted power laws and provide model constants for the full-star model, which then provides excellent agreement with the full-star simulation. The importance of the different components in the model is compared, and are all shown to be relevant. An examination of the effect of initial conditions was then conducted using the one-dimensional model, which would have been infeasible in three dimensions. More mass was burned when the ignition kernel was larger and closer to the center of the star. The turbulent flame speed was found to be important during the early-time evolution of the thermal, but played a diminished role at later times when the evolution is dominated by the large-scale hydrodynamics responsible for entrainment. However, a higher flame speed effectively gave a larger initial ignition kernel and so resulted in more mass burned. This suggests that future studies should focus on the early-time behavior of these thermals (in particular, the transition to turbulence), and that the choice of turbulent flame speed does not play a significant role in the dynamics once the thermal has become established.

ON THE USE OF HIGHER-ORDER PROJECTION METHODS FOR INCOMPRESSIBLE TURBULENT FLOW ∗

An important issue in the development of higher-order methods for incompressible flow is how they perform when the flow is turbulent. A useful diagnostic of a method for turbulent flow is the minimum resolution that is required to adequately resolve the turbulent energy cascade at a given Reynolds number. In this paper, we present careful numerical experiments to assess the utility of higher-order numerical methods based on this metric. We first introduce a numerical method for the incompressible Navier--Stokes equations based on fourth-order discretizations in both space and time. The method is based on an auxiliary variable formulation and combines fourth-order finite volume differencing with a semi-implicit spectral deferred correction temporal integration scheme. We also introduce, for comparison purposes, versions based on second-order spatial and/or temporal discretizations. We demonstrate that for smooth problems, each of the methods exhibits the expected order of convergence in time and space. We ...

Turbulent oxygen flames in type Ia supernovae

In previous studies, we examined turbulence-flame interactions in carbon-burning thermonuclear flames in Type Ia supernovae. In this study, we consider turbulence-flame interactions in the trailing oxygen flames. The two aims of the paper are to examine the response of the inductive oxygen flame to intense levels of turbulence, and to explore the possibility of transition to detonation in the oxygen flame. Scaling arguments analogous to the carbon flames are presented and then compared against three-dimensional simulations for a range of Damkohler numbers (Da16) at a fixed Karlovitz number. The simulations suggest that turbulence does not significantly affect the oxygen flame when Da16 1, turbulence enhances heat transfer and drives the propagation of a flame that is narrower than the corresponding inductive flame would be. Furthermore, burning under these conditions appears to occur as part of a combined carbon-oxygen turbulent flame with complex compound structure. The simulations do not appear to support the possibility of a transition to detonation in the oxygen flame, but do not preclude it either.

Type Ia supernovae: Advances in large scale simulation

There are two principal scientific objectives in the study of Type Ia supernovae - first, a better understanding of these complex explosions from as near first principles as possible, and second, enabling the more accurate utilization of their emission to measure distances in cosmology. Both tasks lend themselves to large scale numerical simulation, yet take us beyond the current frontiers in astrophysics, combustion science, and radiation transport. Their study requires novel approaches and the creation of new, highly scalable codes.