Bruno Denet

Model Equation for the Dynamics of Wrinkled Shockwaves: Comparison with DNS and Experiments

A model equation for the dynamics and the geometry of the wrinkled front of shock waves, obtained for strong shocks in the Newtonian limit, is tested by comparison with direct numerical simulations and a shock tube experiment.

Darrieus–Landau instability of premixed flames enhanced by fuel droplets

Recent experiments on spray flames propagating in a Wilson cloud chamber have established that spray flames are much more sensitive to wrinkles or corrugations than single-phase flames. To propose certain elements of explanation, we numerically study the Darrieus–Landau (or hydrodynamic) instability (DL-instability) developing in premixtures that contain an array of fuel droplets. Two approaches are compared: numerical simulation starting from the general conservation laws in reactive media, and the numerical computation of Sivashinsky-type model equations for DL-instability. Both approaches provide us with results in deep agreement. It is first shown that the presence of droplets in fuel–air premixtures induces initial perturbations which are large enough to trigger the DL-instability. Second, the droplets are responsible for additional wrinkles when the DL-instability is developed. The latter wrinkles are of length scales shorter than those of the DL-instability, in such a way that the DL-unstable spray flames have a larger front surface and therefore propagate faster than the single-phase ones when subjected to the same instability.

Supernovae: An example of complexity in the physics of compressible fluids

Because the collapse of massive stars occurs in a few seconds, while the stars evolve on billions of years, the supernovae are typical complex phenomena in fluid mechanics with multiple time scales. We describe them in the light of catastrophe theory, assuming that successive equilibria between pressure and gravity present a saddle-center bifurcation. In the early stage we show that the loss of equilibrium may be described by a generic equation of the Painlevé I form. This is confirmed by two approaches, first by the full numerical solutions of the Euler-Poisson equations for a particular pressure-density relation, secondly by a derivation of the normal form of the solutions close to the saddle-center. In the final stage of the collapse, just before the divergence of the central density, we show that the existence of a self-similar collapsing solution compatible with the numerical observations imposes that the gravity forces are stronger than the pressure ones. This situation differs drastically in its principle from the one generally admitted where pressure and gravity forces are assumed to be of the same order. Moreover it leads to different scaling laws for the density and the velocity of the collapsing material. The new self-similar solution (based on the hypothesis of dominant gravity forces) which matches the smooth solution of the outer core solution, agrees globally well with our numerical results, except a delay in the very central part of the star, as discussed. Whereas some differences with the earlier self-similar solutions are minor, others are very important. For example, we find that the velocity field becomes singular at the collapse time, diverging at the center, and decreasing slowly outside the core, whereas previous works described a finite velocity field in the core which tends to a supersonic constant value at large distances. This discrepancy should be important for explaining the emission of remnants in the post-collapse regime. Finally we describe the post-collapse dynamics, when mass begins to accumulate in the center, also within the hypothesis that gravity forces are dominant.Graphical abstract

Flame wrinkles from the Zhdanov–Trubnikov equation

Abstract The Zhdanov–Trubnikov equation describing wrinkled premixed flames is studied, using pole decompositions as starting points. Its one-parameter ( − 1 ⩽ c ⩽ + 1 ) nonlinearity generalises the Michelson–Sivashinsky equation ( c = 0 ) to a stronger Darrieus–Landau instability. The shapes of steady flame crests (or periodic cells) are deduced from Laguerre (or Jacobi) polynomials when c ≈ − 1 , which numerical resolutions confirm. Large wrinkles are analysed via a pole density: adapting results of Dunkl relates their shapes to the generating function of Meixner–Pollaczek polynomials, which numerical results confirm for − 1 c ⩽ 0 (reduced stabilisation). Although locally ill-behaved if c > 0 (over-stabilisation) such analytical solutions can yield accurate flame shapes for 0 ⩽ c ⩽ 0.6 . Open problems are invoked.

Premixed flame dynamics in presence of mist

ABSTRACT The injection of a water spray within an enclosure prone to explosion is reputed to reduce the risk. This strategy for safety improvement is at the root of numerous experiments that have concluded that premixed flame can be extinguished by a sufficient amount of a water aerosol characterized by suitable droplet sizes. On the other hand, certain experiments seemingly indicate that flame speed promotion can be observed when particular water mists are injected within the pre-mixture. To contribute to shed light upon these less than intuitive observations, we propose to study the propagation of a nearly stoichiometric premixed flame within a 2D-lattice of water droplets. Main parameters of investigation are droplet size and droplet inter-distance (or equivalently, lattice spacing). When the droplet inter-distance is small, the results confirm that a sufficient amount of water quenches combustion. For larger droplet inter-distance, we observe a flame speed enhancement for suitable droplet size. Concomitantly, the flame front folds subjected to Darrieus–Landau (DL) instability. The final discussion, which invokes a Sivashinsky-type model equation for DL instability, interprets such a speed promotion in presence of mist as a secondary nonlinear enhancement of the flame surface.

Premixed-flame shapes and polynomials

Abstract The nonlinear nonlocal Michelson–Sivashinsky equation for isolated crests of unstable flames is studied, using pole-decompositions as starting point. Polynomials encoding the numerically computed 2 N flame-slope poles, and auxiliary ones, are found to closely follow a Meixner–Pollaczek recurrence; accurate steady crest shapes ensue for N ≥ 3 . Squeezed crests ruled by a discretized Burgers equation involve the same polynomials. Such explicit approximate shapes still lack for finite- N pole-decomposed periodic flames, despite another empirical recurrence.

Shapes and speeds of steady forced premixed flames

Steady premixed flames subjected to space-periodic steady forcing are studied via inhomogeneous Michelson-Sivashinsky (MS) and then Burgers equations. For both, the flame slope is posited to comprise contributions from complex poles to locate, and from a base-slope profile chosen in three classes (pairs of cotangents, single-sine functions or sums thereof). Base-slope-dependent equations for the pole locations, along with formal expressions for the wrinkling-induced flame-speed increment and the forcing function, are obtained on excluding movable singularities from the latter. Besides exact few-pole cases, integral equations that rule the pole density for large wrinkles are solved analytically. Closed-form flame-slope and forcing-function profiles ensue, along with flame-speed increment vs forcing-intensity curves; numerical checks are provided. The Darrieus-Landau instability mechanism allows MS flame speeds to initially grow with forcing intensity much faster than those of identically forced Burgers fronts; only the fractional difference in speed increments slowly decays at intense forcing, which numerical (spectral) timewise integrations also confirm. Generalizations and open problems are evoked.