arXiv subjects

Bruno Denet

Publications and source records attributed to Bruno Denet.

15 recordsLinked to original sources

arXiv2019

Nonlinear response of wrinkled premixed flames to time- and space-dependent forcing and stretch

Premixed-flame wrinkling is studied via a Michelson-Sivashinsky (MS) type of evolution equation retaining the Darrieus-Landau (DL) instability, a curvature effect and a geometric nonlinearity. Here it also keeps forcing by longitudinal shearflow and wrinkle stretch by transverse flow; both imposed stimuli vary in time and space as to make the front slope comprise a given fluctuating spatial harmonics and unknown pole-decomposed pieces. A DL-free Burgers version is examined in parallel, also with Neumann conditions and symmetry. As is shown for both models, solving Ntot equations of motion for the poles in principle yields the front dynamics, the arclength increment V(t) and its time-average Vav. Yet this could be worked out analytically (or nearly so) only in high-frequency HF or low-frequency LF limits. These tackle one or two pairs of poles per cell, then a large number of pairs Ntot forming two piles viewed as continua, one per crest. Despite ample pole motions that make some commute between crests, Vav grows in a nearly parabolic way with the combined intensity of forcing and stretch. LF stimuli and DL instability can induce multiple branches and relaxation phenomena. Numerical t-averages are needed even if V(t) is analytically known. For Ntot=1,2 and short wrinkles, or Burgers fronts, Vav transitions from quadratic to sublinear as the forcing grows ; for longer wrinkles Vav keeps its MS value at moderate forcing, then bifurcates to an ultimately sublinear growth that depends on the stimulus phases. For very long wrinkles, coupled integral equations give analytical slope and pole-density profiles, but pile heights/contents need a t-dependent numerical search of up to two roots to get V(t). A summary, a discussion and hints of generalizations are provided, and open problems are evoked.

nlin.PS↗

arXiv2019

Supernova implosion-explosion in the light of catastrophe theory

The present understanding of supernova explosion of massive stars as a two-step process, with an initial gravitational collapse toward the center of the star followed by an expansion of matter after a bouncing on the core, meets several difficulties. We show that it is not the only possible one: a simple model based on fluid mechanics, catastrophe theory, and stability properties of the equilibrium state shows that one can have also a simultaneous inward/outward motion in the early stage of the instability of the supernova described by a dynamical saddle-center bifurcation. The existence of this simultaneous inward/outward motion is sensitive to the model in such systems with long-range interactions. If a constant temperature is assumed (canonical ensemble), an overall inward motion occurs, but if one imposes with the same equation of state the constraint of energy conservation (microcanonical ensemble) there is an inward velocity field near the center of the star together with an outward velocity field in the rest of the star. We discuss the expansion stage of the remnants away from the collapsed core, and propose a new explanation for the formation of shock waves in the ejecta which differs from the usual Sedov-Taylor self-similar description.

astro-ph.SR↗

arXiv2019

Implosion-explosion in supernovae

Supernovae explosions of massive stars are nowadays believed to result from a two-step process, with an initial gravitational core collapse followed by an expansion of matter after a bouncing on the core. This scenario meets several difficulties. We show that it is not the only possible one: a simple model based on fluid mechanics and stability properties of the equilibrium state shows that one can have also a simultaneous inward/outward motion in the early stage of the instability of the supernova. This shows up in the slow sweeping across a saddle-center bifurcation found when considering equilibrium states associated to the constraint of energy conservation. We first discuss the weakly nonlinear regime in terms of a Painlev\'e I equation. We then show that the strongly nonlinear regime displays a self-similar behavior of the core collapse. Finally, the expansion of the remnants is revisited as an isentropic process leading to shocks formation.

astro-ph.SR↗

arXiv2014

Premixed flame shapes and polynomials

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 2N 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 shape still lack for finite-N pole-decomposed periodic flames, despite another empirical recurrence.

nlin.PS↗

arXiv2014

Shapes and speeds of 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.

nlin.PS↗

arXiv2013

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

The supernovae are typical complex phenomena in fluid mechanics with very different time scales. We describe them in the light of catastrophe theory, assuming that successive equilibria between pressure and gravity present a saddle-node bifurcation. In the early stage we show that the loss of equilibrium may be described by a generic equation of the Painlev\'e I form. 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. Our 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. 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.

physics.flu-dyn↗

arXiv2013

Resolvent methods for steady premixed flame shapes governed by the Zhdanov-Trubnikov equation

Using pole decompositions as starting points, the one parameter (-1 =< c < 1) nonlocal and nonlinear Zhdanov-Trubnikov (ZT) equation for the steady shapes of premixed gaseous flames is studied in the large-wrinkle limit. The singular integral equations for pole densities are closely related to those satisfied by the spectral density in the O(n) matrix model, with n = -2(1 + c)/(1 - c). They can be solved via the introduction of complex resolvents and the use of complex analysis. We retrieve results obtained recently for -1 =< c =< 0, and we explain and cure their pathologies when they are continued naively to 0 < c < 1. Moreover, for any -1 =< c < 1, we derive closed-form expressions for the shapes of steady isolated flame crests, and then bicoalesced periodic fronts. These theoretical results fully agree with numerical resolutions. Open problems are evoked.

physics.flu-dyn↗

arXiv2012

Flame Wrinkles From the Zhdanov-Trubnikov Equation

The Zhdanov-Trubnikov equation describing wrinkled premixed flames is studied, using pole-decompositions as starting points. Its one-parameter (-1< c <1) nonlinearity generalizes 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 0 (over-stabilization) such analytical solutions can yield accurate flame shapes for 0< c <0.6. Open problems are invoked.

nlin.PS↗

arXiv2011

Wrinkled flames and geometrical stretch

Localized wrinkles of thin premixed flames subject to hydrodynamic instability and geometrical stretch of uniform intensity (S) are studied. A stretch-affected nonlinear and nonlocal equation, derived from an inhomogeneous Michelson-Sivashinsky equation, is used as a starting point, and pole decompositions are used as a tool. Analytical and numerical descriptions of isolated (centered or multicrested) wrinkles with steady shapes (in a frame) and various amplitudes are provided; their number increases rapidly with 1/S > 0. A large constantS > 0 weakens or suppresses all localized wrinkles (the larger the wrinkles, the easier the suppression), whereasS < 0 strengthens them; oscillations of S further restrict their existence domain. Self-similar evolutions of unstable many-crested patterns are obtained. A link between stretch, nonlinearity, and instability with the cutoff size of the wrinkles in turbulent flames is suggested. Open problems are evoked.

physics.flu-dyn↗

arXiv2008

The Sivashinsky equation for corrugated flames in the large-wrinkle limit

Sivashinsky's (1977) nonlinear integro-differential equation for the shape of corrugated 1-dimensional flames is ultimately reducible to a 2N-body problem, involving the 2N complex poles of the flame slope. Thual, Frisch & Henon (1985) derived singular linear integral equations for the pole density in the limit of large steady wrinkles $(N \gg 1)$, which they solved exactly for monocoalesced periodic fronts of highest amplitude of wrinkling and approximately otherwise. Here we solve those analytically for isolated crests, next for monocoalesced then bicoalesced periodic flame patterns, whatever the (large-) amplitudes involved. We compare the analytically predicted pole densities and flame shapes to numerical results deduced from the pole-decomposition approach. Good agreement is obtained, even for moderately large Ns. The results are extended to give hints as to the dynamics of supplementary poles. Open problems are evoked.

physics.class-ph↗

arXiv2007

Sivashinsky equation in a rectangular domain

The (Michelson) Sivashinsky equation of premixed flames is studied in a rectangular domain in two dimensions. A huge number of 2D stationary solutions are trivially obtained by addition of two 1D solutions. With Neumann boundary conditions, it is shown numerically that adding two stable 1D solutions leads to a 2D stable solution. This type of solution is shown to play an important role in the dynamics of the equation with additive noise.

physics.class-ph↗

arXiv2006

Potential model of a 2D Bunsen flame

The Michelson Sivashinsky equation, which models the non linear dynamics of premixed flames, has been recently extended to describe oblique flames. This approach was extremely successful to describe the behavior on one side of the flame, but some qualitative effects involving the interaction of both sides of the front were left unexplained. We use here a potential flow model, first introduced by Frankel, to study numerically this configuration. Furthermore, this approach allows us to provide a physical explanation of the phenomena occuring in this geometry by means of an electrostatic analogy.

physics.class-ph↗

arXiv2006

Non-linear model equation for three-dimensional Bunsen flames

The non linear description of laminar premixed flames has been very successful, because of the existence of model equations describing the dynamics of these flames. The Michelson Sivashinsky equation is the most well known of these equations, and has been used in different geometries, including three-dimensional quasi-planar and spherical flames. Another interesting model, usually known as the Frankel equation,which could in principle take into account large deviations of the flame front, has been used for the moment only for two-dimensional expanding and Bunsen flames. We report here for the first time numerical solutions of this equation for three-dimensional flames.

physics.class-ph↗

arXiv2006

Stationary solutions and Neumann boundary conditions in the Sivashinsky equation

New stationary solutions of the (Michelson) Sivashinsky equation of premixed flames are obtained numerically in this paper. Some of these solutions, of the bicoalescent type recently described by Guidi and Marchetti, are stable with Neumann boundary conditions. With these boundary conditions, the time evolution of the Sivashinsky equation in the presence of a moderate white noise is controlled by jumps between stationary solutions.

physics.class-ph↗

arXiv2005

Low vorticity and small gas expansion in premixed flames

Different approaches to the nonlinear dynamics of premixed flames exist in the literature: equations based on developments in a gas ex- pansion parameter, weak nonlinearity approximation, potential model equation in a coordinate-free form. However the relation between these different equations is often unclear. Starting here with the low vor- ticity approximation proposed recently by one of the authors, we are able to recover from this formulation the dynamical equations usually obtained at the lowest orders in gas expansion for plane on average flames, as well as obtain a new second order coordinate-free equation extending the potential flow model known as the Frankel equation. It is also common to modify gas expansion theories into phenomelogical equations, which agree quantitatively better with numerical simula- tions. We discuss here what are the restrictions imposed by the gas expansion development results on this process.

physics.class-ph↗