Pierre Haldenwang

Spectral and finite difference solutions of the Burgers equation

Abstract Spectral methods (Fourier Galerkin, Fourier pseudospectral, Chebyshev Tau, Chebyshev collocation, spectral element) and standard finite differences are applied to solve the Burgers equation with small viscosity ( v = 1 100 π ). This equation admits a (nonsingular) thin internal layer that must be resolved if accurate numerical solutions are to be obtained. From the reported computations, it appears that spectral schemes offer the best accuracy, especially if coordinate transformation or elemental subdivision is used to resolve the regions of large variation of the dependent variable.

A Numerical Study of Premixed Flames Darrieus-Landau Instability

The complete equations of premixed flames are solved numerically, in the isobaric approximation and with a simplified chemical kinetics. A momentum-pressure formulation is proposed for solving non-constant density flows. The growth rates of the Darrieus-Landau instability are measured and compared to the linear theory. Large amplitude curved flames are obtained, as well as flames submitted to a shear flow.

Numerical Study of Thermal-Diffusive Instability of Premixed Flames

Abstract The 2-D thennal-diffusive model of premixed flames is solved numerically. The growth rates of the thermal-diffusive instability are compared to the linear theory. It is shown that the discrepancy, although large (a relative error than can be larger than 100%), behaves like O(I/β) as expected by asymptotics (β being the reduced activation energy or Zeldovich number). We additionally present results far in the non-linear domain. They exhibit turbulent behaviour which are qualitatively similar to the dynamical properties of the Kuramoto-Sivashinsky model-equation.

Supercritical burning of liquid oxygen (LOX) droplet with detailed chemistry

Abstract A numerical study of the supercritical combustion of a liquid oxygen (LOX) droplet in a stagnant environment of hot hydrogen is carried out with a detailed chemistry model. Special attention is devoted to ignition process and diffusion flame structure. Ignition consists typically of the propagation of a premixed flame which is initiated in the H 2 -rich hot side. Propagation takes place in a nonhomogeneous hot environment (say 1500 K) with a considerable velocity (typically 50 ms −1 ). Despite the high temperature of the ambiance, the medium ahead of the flame can be considered as frozen during the transit time. In addition, it is found that droplets with diameter less than 1 μm vaporize before burning. A quasi-steady-like diffusion flame is then established. In this regime we observe that the D 2 law is approximately valid. In contrast to the case of a single irreversible reaction, a full chemistry model leads to a very thick flame where chemical consumption and production cover a surrounding zone about four times the instantaneous droplet radius. Reversibility of the reactions plays a determinant role in the flame structure by inducing a large near-equilibrium zone which is separated from a frozen region by a thin nonequilibrium zone. The length scale of the latter region is found to behave as the square root of the instantaneous droplet radius and a detailed analysis shows that just two elementary reactions are involved in this zone. Furthermore, the influence of several parameters is considered; temperature and pressure in the combustion chamber have a weak influence on the burning time. Influence of initial droplet radius confirms that droplet combustion is a diffusion controlled process. Chamber composition is also considered. Finally, it is shown that a precise description of the transport properties in the dense phase is not required.

High pressure vaporization of LOX droplet crossing the critical conditions

Abstract Vaporization of liquid O2 droplet in quiescent high-temperature and high-pressure H2 gas is numerically investigated. Classical thermodynamic modeling of high pressure mixtures allows us to study the transition from subcritical to supercritical vaporization regime. It is observed that subcritical vaporization can be obtained up to pressures several times the oxygen critical pressure. Respective domain of both regimes is determined vs temperature and pressure. Border region corresponds to minimum value of droplet lifetime. This results from two cooperative phenomena: transient effect and thermodynamic property of mixtures. Sensitivity analysis additionally shows that state of art in dense fluid transport modeling yields results that should be considered accurate only as far as orders of magnitude are concerned.

High order scheme for thermally driven flows in an open channel

Abstract The paper presents a high order numerical scheme for solving thermally driven flows in an open duct. More precisely, this approach deals with flows at large Rayleigh number and large aspect ratio of channel (length≫spacing). For such flows we propose an appropriate set of boundary conditions which is devoted to approach the part played by the return flow that reconnects outlet to inlet in experiments and consequently that allows us to focus on the channel itself. Additional effort has been made for implementing a high order scheme in order to achieve accuracy. A 2-D mixed scheme is presented: Chebyshev collocation method in the direction transverse to the flow and fourth order finite difference schemes in the streamwise direction, i.e. parallel to the vertical plates. The scheme accuracy is shortly checked vs an elliptic problem and with respect to a classical benchmark for numerical studies of free convection: the thermally driven cavity. Afterwards, we consider the set of boundary conditions that leads to fulfil a satisfactory comparison with two experiments of free convective flows in a vertical open channel. Both cases correspond to well-documented experiments of natural convection in thermosiphon. The first comparison considers the thermal transfer due to the flow induced by symmetrical wall heating at uniform heat flux. Our scheme predicts it with 2% accuracy on a large range of parameters. Secondly, we present a successful numerical attempt concerning the phenomenon of flow reversal which appears in the case of non-symmetrical heating. The computed threshold of its onset differs from experimental observations by about 10%.

A local extinction of the thermo-diffusive premixed flame at low Lewis number

We solve the 2-D thermo-diffusive model of premixed flames in the framework of Fourier Spectral Methods. Although the temperature and concentration fields are not periodic in the direction perpendicular to the flame, we suggest a particular treatment, simple to implement, that is applied to these quantities in order to transform them into the sum of a known profile and a periodic unknown. This process also takes advantage of the fact that the physics of flames allows us to consider as periodic the higher derivatives. “Infinite” order convergence of Spectral Methods is thus recovered. This algorithm being very efficient, we can perform numerical simulation concerning the diffusive-thermal instability, far in the non-linear domain. Thus, at low Lewis number, we numerically observed, for the first time to our knowledge, a phenomenon of local extinction. This brings a plausible explanation to the presence of unburnt combustible in the lean hydrogen flame.