Mach Number Expansion Research Articles

Evolution of Rayleigh streaming flow velocity components in a resonant waveguide at high acoustic levels

The interaction between an acoustic wave and a solid wall generates a mean steady flow called Rayleigh streaming, generally assumed to be second order in a Mach number expansion. This flow is well known in the case of a stationary plane wave at low amplitude: it has a half-wavelength spatial periodicity and the maximum axial streaming velocity is a quadratic function of the acoustic velocity amplitude at the antinode. For higher acoustic levels, additional streaming cells have been observed. In the present study, results of LDV and PIV measurements are compared to direct numerical simulations. The evolution of axial and radial velocity components for both acoustic and streaming flows is studied from low to high acoustic amplitudes. Two streaming flow regimes are pointed out, the axial streaming dependency upon acoustics going from quadratic to linear. The hypothesis of the radial streaming velocity being of second order in a Mach number expansion is shown to be invalid at high amplitudes. The change of regime occurs when the radial streaming velocity amplitude becomes larger than the radial acoustic velocity amplitude, high levels being therefore characterized by nonlinear interaction of the different velocity components.

Acoustic and streaming velocity components in a resonant waveguide at high acoustic levels.

Rayleigh streaming is a steady flow generated by the interaction between an acoustic wave and a solid wall, generally assumed to be second order in a Mach number expansion. Acoustic streaming is well known in the case of a stationary plane wave at low amplitude: it has a half-wavelength spatial periodicity and the maximum axial streaming velocity is a quadratic function of the acoustic velocity amplitude at antinode. For higher acoustic levels, additional streaming cells have been observed. Results of laser Doppler velocimetry measurements are here compared to direct numerical simulations. The evolution of axial and radial velocity components for both acoustic and streaming velocities is studied from low to high acoustic amplitudes. Two streaming flow regimes are pointed out, the axial streaming dependency on acoustics going from quadratic to linear. The evolution of streaming flow is different for outer cells and for inner cells. Also, the hypothesis of radial streaming velocity being of second order in a Mach number expansion, is not valid at high amplitudes. The change of regime occurs when the radial streaming velocity amplitude becomes larger than the radial acoustic velocity amplitude, high levels being therefore characterized by nonlinear interaction of the different velocity components.

Asymptotic theory of a uniform flow of a rarefied gas past a sphere at low Mach numbers

A slow uniform flow of a rarefied gas past a sphere with a uniform temperature is considered. The steady behaviour of the gas is investigated on the basis of the Boltzmann equation by a systematic asymptotic analysis for small Mach numbers in the case where the Knudsen number is finite. Introducing a slowly varying solution whose length scale of variation is much larger than the sphere dimension, the fluid-dynamic-type equations describing the overall behaviour of the gas in the far region are derived. Then, the solution in the near region which varies on the scale of the sphere size, described by the linearised Boltzmann equation, and the solution in the far region, described by the fluid-dynamic-type equations, are sought in the form of a Mach number expansion up to the second order, in a way that they are joined in the intermediate overlapping region. As a result, the drag is derived up to the second order of the Mach number, which formally extends the linear drag obtained by Takata et al. (Phys. Fluids A, vol. 5, 1993, pp. 716–737) to a weakly nonlinear case. Numerical results for the drag on the basis of the Bhatnagar–Gross–Krook (BGK) model are also presented.

Influence of gas compression on flame acceleration in the early stage of burning in tubes

The mechanism of finger flame acceleration at the early stage of burning in tubes was studied experimentally by Clanet and Searby [Combust. Flame 105 (1996) 225] for slow propane-air flames, and elucidated analytically and computationally by Bychkov et al. [Combust. Flame 150 (2007) 263] in the limit of incompressible flow. We have now analytically, experimentally and computationally studied the finger flame acceleration for fast burning flames, when the gas compressibility assumes an important role. Specifically, we have first developed a theory through small Mach number expansion up to the first-order terms, demonstrating that gas compression reduces the acceleration rate and the maximum flame tip velocity, and thereby moderates the finger flame acceleration noticeably. This is an important quantitative correction to previous theoretical analysis. We have also conducted experiments for hydrogen–oxygen mixtures with considerable initial values of the Mach number, showing finger flame acceleration with the acceleration rate much smaller than those obtained previously for hydrocarbon flames. Furthermore, we have performed numerical simulations for a wide range of initial laminar flame velocities, with the results substantiating the experiments. It is shown that the theory is in good quantitative agreement with numerical simulations for small gas compression (small initial flame velocities). Similar to previous works, the numerical simulation shows that finger flame acceleration is followed by the formation of the “tulip” flame, which indicates termination of the early acceleration process.

Flow effects on the acoustic end correction of a sudden in-duct area expansion

For scattering of plane waves at a sudden area expansion in a duct, the presence of flow may significantly alter the reactive properties. This paper studies the influence of a mean flow field and unstable separated flow on the reactive properties of the expansion, formulated as an end correction. Theoretical and experimental results show that the expansion end correction is significantly affected by the flow and hydrodynamic waves excited at the edge of the expansion. The effects are different in three regions where the Strouhal number is small, of order 1, and large. The influence is most significant at Strouhal numbers of the order 1, with specific limiting values for large and small Strouhal numbers, respectively. In the analytic model, an important feature is the shear layer at the edge modeled as a vortex sheet with the unsteady Kutta condition applied at the edge. The influence of Mach number, Helmholtz number, and area expansion ratio is studied, and a quasistationary, small Strouhal number, approximation yields an expression for the end correction. Further, the influence of edge condition is explored, emphasizing the importance of interaction between sound and unsteady vorticity shedding at the edge of the area expansion.

Aeroacoustic Simulation around a Circular Cylinder by the Equations Split for Incompressible Flow Field and Acoustic Field

A method for computational aeroacoustics using the equations split into the incompressible flow part and the acoustic part is investigated in this paper. The equations derived based on Mach number expansion, and the method to solve them with high accuracy and stability are shown. The finite volume method using WENO scheme is applied to the acoustic field around a circular cylinder. Comparing the present method with the approximate solution based on the acoustic analogy, it is found that the acoustic field has been decayed according to the square root of the distance from the object further from about 10 times the diameter. In addition, 15 grid points per a wave length can give a sufficiently accurate solution by the present method.

Influence of compressibility on scaling regimes of strongly anisotropic fully developed turbulence

A statistical model of strongly anisotropic fully developed turbulence of the weakly compressible fluid is considered by means of the field theoretic renormalization group. The corrections due to compressibility to the infrared form of the kinetic energy spectrum have been calculated in the leading order in the Mach number expansion. Furthermore, in this approximation the validity of the Kolmogorov hypothesis on the independence of the dissipation length of velocity correlation functions in the inertial range has been proved.

Renormalization group, operator product expansion, and anomalous scaling in a model of advected passive scalar

Statistical model of strongly anisotropic fully developed turbulence of the weakly compressible fluid is considered by means of the field theoretic renormalization group. The corrections due to compressibility to the infrared form of the kinetic energy spectrum have been calculated in the leading order in Mach number expansion. Furthermore, in this approximation the validity of the Kolmogorov hypothesis on the independence of dissipation length of velocity correlation functions in the inertial range has been proved.

Quenching of reaction in gas-sampling probes to measure scramjet engine performance

The problem of chemical quenching in gas-sampling probes was studied as a reactive, Rayleigh, and Fanno flow. The typical lengths of friction, convective heat transfer, and reactions in the probes are evalnated under the testing conditions of ramjet and scramjet engines. A reduced chemical kinetic scheme was adopted, and the reaction times were evaluated from the ignition times derived by reflected shock wave experiments. Thereby, criteria for quenching in probes were obtained analytically. The critical Damköhler number ( Da ) for quenching in an H 2 -air reaction was found to be 23 (1500 K) to 10.5 (3000 K). A 2.6-fold greater Da was derived for CH 4 -air mixtures. Because reactions in CH 4 -air mixtures can be quenched even above 2200 K, combustion performance in a CH 4 -fueled ramjet combustor was measured from gas analysis. However, quenching of H 2 unburned gas in more difficult than quenching of CH 4 , Cooling with supersonic expansion is essential to freeze reactions in H 2 -air mixtures in scramjets tested under a total temperature higher than 2000 K. A constant Mach number expansion is recommended downstream of the sampling orifice. The contour of the expansion region and the decay of total pressure were determined taking into account friction and cooling effects. Reactions with unburned H 2 can be quenched if the sampling probe works as a supersonic diffuser operating in the supercritical mode.

Perturbation solutions for variable energy blast waves

The trajectory of and the flow field behind blast waves with time varying energy input is determined. Freeman's (1968) Lagrangean coordinate formulation is modified to include both the geometric factor, α, for plane, cylindrical and spherical shocks and also non-integer values of β, the energy input parameter, in a single computational algorithm. Numerical problems associated with vanishing density at the inner mass boundary or “piston face” are then examined and solved. Second order perturbation solutions about the solution for an infinite strength shock are then obtained in Sakurai's (1965) inverse shock Mach number expansion parameter for 0 ⩽ β < α + 1. Tables and graphs of significant numerical coefficients are presented for comparison to, and extension of, results of other authors. Graphs of typical shock trajectories and flow field density, pressure and velocity variations are also presented and discussed.

On non-uniform mach number expansion of the Navier-Stokes equations and its relation to aerodynamically generated sound

This paper is concerned with a study of the main effects of incorporating the compressibility into the Navier-Stokes equations in the non-stationary case. The results that are presented seem quite contrary to arguments common to the literature on subsonic Mach number expansions in the manner in which the perturbation technique is subject to modification by its susceptibility to the time dependence of the problem. It is shown that a proper treatment of a time dependent Mach number expansion inevitably requires an admission of a singularity in the perturbation sequence directly attributable to the sound field generated by the flow. This finding has interesting corollaries in the related studies of aerodynamic noise by suggesting an approach to this intriguing subject which does not appear to have been used before.