We present a theoretical stability analysis for an expanding accretion shock that does not involve a rarefaction wave behind it. The dispersion equation that determines the eigenvalues of the problem and the explicit formulae for the corresponding eigenfunction profiles are presented for an arbitrary equation of state and finite-strength shocks. For spherically and cylindrically expanding steady shock waves, we demonstrate the possibility of instability in a literal sense, a power-law growth of shock-front perturbations with time, in the range of $h_c< h<1+2 {\mathcal {M}}_2$ , where $h$ is the D'yakov-Kontorovich parameter, $h_c$ is its critical value corresponding to the onset of the instability and ${\mathcal {M}}_2$ is the downstream Mach number. Shock divergence is a stabilizing factor and, therefore, instability is found for high angular mode numbers. As the parameter $h$ increases from $h_c$ to $1+2 {\mathcal {M}}_2$ , the instability power index grows from zero to infinity. This result contrasts with the classic theory applicable to planar isolated shocks, which predicts spontaneous acoustic emission associated with constant-amplitude oscillations of the perturbed shock in the range $h_c< h<1+2 {\mathcal {M}}_2$ . Examples are given for three different equations of state: ideal gas, van der Waals gas and three-terms constitutive equation for simple metals.
Linear interaction analysis (LIA) is employed to investigate the interaction of reactive and nonreactive shock waves with isotropic vortical turbulence. The analysis is carried out, through Laplace-transform technique, accounting for long-time effects of vortical disturbances on the burnt-gas flow in the fast-reaction limit, where the reaction-region thickness is significantly small in comparison with the most representative turbulent length scales. Results provided by the opposite slow-reaction limit are also recollected. The reactive case is here restricted to situations where the overdriven detonation front does not exhibit self-induced oscillations nor inherent instabilities. The interaction of the planar detonation with a monochromatic pattern of perturbations is addressed first, and then a Fourier superposition for three-dimensional isotropic turbulent fields is employed to provide integral formulas for the amplification of the kinetic energy, enstrophy, and anisotropy downstream. Transitory evolution is also provided for single-frequency disturbances. In addition, further effects associated to the reaction rate, which have not been included in LIA, are studied through direct numerical simulations. The numerical computations, based on WENO-BO4-type scheme, provide spatial profiles of the turbulent structures downstream for four different conditions that include nonreacting shock waves, unstable reacting shock (sufficiently high activation energy), and stable reacting shocks for different detonation thicknesses. Effects of the propagation Mach number, chemical heat release, and burn rate are analyzed.
Very lean hydrogen flames were thought to quench in narrow confined geometries. We show for the first time how flames with very low fuel concentration undergo an unprecedented propagation in narrow gaps: H 2-air flames can survive very adverse conditions by breaking the reaction front into isolated flame cells that travel steadily in straight lines or split to perform a fractal-like propagation that resembles the pathway of starving fungi or bacteria. The combined effect of hydrogen mass diffusivity and intense heat losses act as the two main mechanisms that explain the experimental observations.
An oblique shock impinging on a shear layer that separates two uniform supersonic streams, of Mach numbers $M_{1}$ and $M_{2}$, at an incident angle $\unicode[STIX]{x1D70E}_{i}$ can produce regular and irregular interactions with the interface. The region of existence of regular shock refractions with stable flow structures is delineated in the parametric space $(M_{1},M_{2},\unicode[STIX]{x1D70E}_{i})$ considering oblique-shock impingement on a supersonic vortex sheet of infinitesimal thickness. It is found that under supercritical conditions, the oblique shock fails to deflect both streams consistently and to provide balanced flow properties downstream. In this circumstance, the flow renders irregular configurations which, in the absence of characteristic length scales, exhibit self-similar pseudosteady behaviours. These cases involve shocks moving upstream at constant speed and increasing their intensity to comply with equilibrium requirements. Differences in the variation of propagation speed among the flows yield pseudosteady configurations that grow linearly with time. Supercritical conditions are described theoretically and reproduced numerically using highly resolved inviscid simulation.
The diffusive strip method (DSM) is a near-exact numerical method for mixing computations initially developed in two dimensions (Meunier & Villermaux, J. Fluid Mech., vol. 662, 2010, pp. 134–172). The method, which consists of following stretched material lines to compute the resulting scalar field a posteriori, is extended here to three-dimensional flows. We describe the procedure and its three-dimensional peculiarity, which relies on the Lagrangian advection of a triangulated surface from which the stretching rate is extracted to infer the scalar field. The method is first validated at moderate Péclet number against a classical pseudospectral method solving the advection–diffusion equation for a Batchelor vortex, and then applied to a simple Taylor–Couette experimental configuration with non-rotating boundary conditions at the top-end disk, bottom-end disk and outer cylinder. This motion, producing an elaborate although controlled steady three-dimensional flow, relies on Ekman pumping arising from the rotation of the inner cylinder. A recurrent two-cell structure is separated by the horizontal mid-plane and formed by stream tubes shaped as nested tori under laminar flow conditions. A scalar blob in the flow experiences a Lagrangian oscillating dynamics undergoing stretchings and compressions, driving the mixing process. The DSM enables the calculation of the blob elongation and scalar concentration distributions through a single variable computation along the advected blob surface, capturing the rich evolution observed in the experiments. Interestingly, the mixing process in this axisymmetric and steady three-dimensional flow leads to a linear growth of surfaces in time similar to the one obtained in a two-dimensional shear. The potentialities, limits and extension of the method to more general flows are finally discussed.
Large-activation-energy asymptotic techniques are used to describe the effects of non-unity Lewis numbers of the fuel on strain-induced extinction of axisymmetric counterflow diffusion flames. The present work extends and clarifies previous investigations by accounting also for variable density and variable transport properties of the gas. In our asymptotic analysis the flame structure near extinction is, at leading order, given by the Burke-Schumann limit of infinitely fast reaction; i.e. two outer regions of equilibrium flow, with the fuel and the oxygen separated by an infinitesimally thin reaction layer where they arrive by diffusion in stoichiometric proportions. The leading-order description provides the basic flow structure, including the flame-sheet location, the fuel-consumption rate, the temperature gradients on both sides of the flame, and the peak value of the temperature, which plays a dominant role in flame extinction and differs significantly from the adiabatic-flame value for non-unity Lewis numbers. In the near-extinction regime small departures, due to finite rates, from the fast-reaction limit are enough to dominate the structure of the reaction layer, and must be taken into account in this thin layer and in the outer chemically frozen regions, where the corrections are associated with the reactants leaking, with small mass fractions, through the flame. The main effect of the differential diffusion in the nearextinction regime is due to the strong modification of the reaction rates resulting from the changes in the Burke-Schumann peak temperature, with only moderate corrections due to leakage of the reactants through the flame. For large values of the overall stoichiometric ratio S of the diffusion flame, defined as the mass of the air stream needed to burn to completion the unit mass of the fuel stream, the extinction conditions occur in a premixed-flame regime, in which the reaction layer is displaced towards the fuel side with respect to the Burke-Schumann flame sheet position and a fraction of the arriving fuel mass flux leaks through the reaction layer, while the mass fraction of the leaking oxygen decreases to negligibly small values. The asymptotic predictions are tested by comparison with numerical integrations of extinction curves based on continuation methods.
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