Avshalom Manela

On the Rayleigh-Bénard problem in the continuum limit

The transition to convection in the Rayleigh–Benard problem at small Knudsen numbers is studied via a linear temporal stability analysis of the compressible “slip-flow” problem. No restrictions are imposed on the magnitudes of temperature difference and compressibility-induced density variations. The dispersion relation is calculated by means of a Chebyshev collocation method. The results indicate that occurrence of instability is limited to small Knudsen numbers (Kn≲0.03) as a result of the combination of the variation with temperature of fluid properties and compressibility effects. Comparison with existing direct simulation Monte Carlo and continuum nonlinear simulations of the corresponding initial-value problem demonstrates that the present results correctly predict the boundaries of the convection domain. The linear analysis thus presents a useful alternative in studying the effects of various parameters on the onset of convection, particularly in the limit of arbitrarily small Knudsen numbers.The transition to convection in the Rayleigh–Benard problem at small Knudsen numbers is studied via a linear temporal stability analysis of the compressible “slip-flow” problem. No restrictions are imposed on the magnitudes of temperature difference and compressibility-induced density variations. The dispersion relation is calculated by means of a Chebyshev collocation method. The results indicate that occurrence of instability is limited to small Knudsen numbers (Kn≲0.03) as a result of the combination of the variation with temperature of fluid properties and compressibility effects. Comparison with existing direct simulation Monte Carlo and continuum nonlinear simulations of the corresponding initial-value problem demonstrates that the present results correctly predict the boundaries of the convection domain. The linear analysis thus presents a useful alternative in studying the effects of various parameters on the onset of convection, particularly in the limit of arbitrarily small Knudsen numbers.

Point vortex model for prediction of sound generated by a wing with flap interacting with a passing vortex.

Acoustic signature of a rigid wing, equipped with a movable downstream flap and interacting with a line vortex, is studied in a two-dimensional low-Mach number flow. The flap is attached to the airfoil via a torsion spring, and the coupled fluid-structure interaction problem is analyzed using thin-airfoil methodology and application of the emended Brown and Michael equation. It is found that incident vortex passage above the airfoil excites flap motion at the system natural frequency, amplified above all other frequencies contained in the forcing vortex. Far-field radiation is analyzed using Powell-Howe analogy, yielding the leading order dipole-type signature of the system. It is shown that direct flap motion has a negligible effect on total sound radiation. The characteristic acoustic signature of the system is dominated by vortex sound, consisting of relatively strong leading and trailing edge interactions of the airfoil with the incident vortex, together with late-time wake sound resulting from induced flap motion. In comparison with the counterpart rigid (non-flapped) configuration, it is found that the flap may act as sound amplifier or absorber, depending on the value of flap-fluid natural frequency. The study complements existing analyses examining sound radiation in static- and detached-flap configurations.

On the acoustic radiation of a pitching airfoil

We examine the acoustic far field of a thin elastic airfoil, immersed in low-Mach non-uniform stream flow, and actuated by small-amplitude sinusoidal pitching motion. The near-field fluid-structure interaction problem is analyzed using potential thin-airfoil theory, combined with a discrete vortex model to describe the evolution of airfoil trailing edge wake. The leading order dipole-sound signature of the system is investigated using Powell-Howe acoustic analogy. Compared with a pitching rigid airfoil, the results demonstrate a two-fold effect of structure elasticity on airfoil acoustic field: at actuation frequencies close to the system least stable eigenfrequency, elasticity amplifies airfoil motion amplitude and associated sound levels; however, at frequencies distant from this eigenfrequency, structure elasticity acts to absorb system kinetic energy and reduce acoustic radiation. In the latter case, and with increasing pitching frequency ωp, a rigid-airfoil setup becomes significantly noisier than an...

On the Rayleigh-Bénard problem : dominant compressibility effects

We study the linear temporal hydrodynamic stability in the Rayleigh-Benard problem for a compressible fluid (a perfect gas) under marginally super-adiabatic conditions, i.e. when the ambient temperature gradient only slightly exceeds the adiabatic gradient and then only within the fluid adjacent to the upper (cold) wall. The onset of convection in this limit demonstrates some unique features which differ qualitatively from those of the familiar Boussinesq approximation. Thus, the ensuing convection is effectively confined to a narrow domain of the fluid close to the upper wall and is characterized by large wavenumbers. Furthermore, these distinct attributes persist with diminishing temperature difference, implying that the prevailing generalized Boussinesq approximation (based on the use of the potential temperature gradient) is non-uniform in the present limit. This non-uniformity is resolved in terms of the small yet significant variations of fluid properties (which are commonly neglected). We comment on the analogy between the present problem and the Taylor-Couette problem for a viscous incompressible fluid within a narrow gap between counter-rotating cylinders. We briefly discuss the potential relevance of the present limit to some recent observations of the onset of convection within near-critical fluids.

On the damping effect of gas rarefaction on propagation of acoustic waves in a microchannel

We consider the response of a gas in a microchannel to instantaneous (small-amplitude) non-periodic motion of its boundaries in the normal direction. The problem is formulated for an ideal monatomic gas using the Bhatnagar, Gross, and Krook (BGK) kinetic model, and solved for the entire range of Knudsen (Kn) numbers. Analysis combines analytical (collisionless and continuum-limit) solutions with numerical (low-variance Monte Carlo and linearized BGK) calculations. Gas flow, driven by motion of the boundaries, consists of a sequence of propagating and reflected pressure waves, decaying in time towards a final equilibrium state. Gas rarefaction is shown to have a “damping effect” on equilibration process, with the time required for equilibrium shortening with increasing Kn. Oscillations in hydrodynamic quantities, characterizing gas response in the continuum limit, vanish in collisionless conditions. The effect of having two moving boundaries, compared to only one considered in previous studies of time-peri...

On the Rayleigh-Bénard problem in the continuum limit: Effects of temperature differences and model of interaction

The transition to convection in the Rayleigh-Benard problem at small Knudsen numbers is studied via a linear temporal stability analysis of the compressible “slip-flow” problem. Considering a power-law (variable hard-sphere) model of interaction our analysis indicates that for sufficiently large Froude numbers “softer” potentials result in less unstable systems. At small Froude numbers this trend is reversed, i.e., the “softer” interaction potentials correspond to a more unstable response. These results are discussed in terms of the opposing mechanisms of thermal expansion and compressibility. We carry out an asymptotic expansion for small temperature differences and establish the principle of exchange of stabilities for this limit. A singularity appears in this limit when compressibility effects are dominant.

Vibration and sound of a flapping airfoil: The limit of small bending rigidity

We investigate the near and far fields of a thin elastic airfoil set at uniform low-Mach flow and subject to leading-edge heaving actuation. The airfoil is “hanged” in the vertical direction and is free at its downstream end, so that “hanging chain” gravity-induced tension forces apply. The structure bending rigidity is assumed small, and we focus on analyzing the differences between a highly elastic airfoil and a membrane (where the bending rigidity vanishes). The near field is studied based on potential thin airfoil theory, whereas the acoustic field is investigated using the Powell-Howe acoustic analogy. The results shed light on the specific effect of structure bending stiffness on the dynamics and acoustic disturbance of an airfoil.

The sound of an elastic airfoil in the wake of a vortex generator

The acoustic signature of an acoustically compact tandem airfoil setup in high-Reynolds number flow is investigated. The upstream airfoil is rigid and is actuated at its leading edge with small-amplitude sinusoidal pitching motion. The downstream airfoil is taken passive and elastic, with its motion forced by the vortex-street excitation of the upstream airfoil. The near-field description is obtained via potential thin-airfoil theory. It is then applied as a source term into the Powell-Howe acoustic analogy, to yield the far-field dipole radiation of the system. To assess the effect of downstream-airfoil elasticity, results are compared with calculations for a non-elastic setup, where the downstream airfoil is rigid and fixed. Depending on separation distance between airfoils, airfoil-motion and airfoil-wake dynamics shift between in-phase (synchronized) and counter-phase behaviors. Downstream airfoil elasticity acts to either amplify or suppress sound, through the direct contribution of elastic-airfoil m...

Thermoacoustic approach for the cloaking of a vibrating surface

A vibrating surface in a quiescent fluid transmits pressure fluctuations, which propagate into far-field sound. The vibroacoustic mechanism involved, coupling vibration and sound, is common in a large number of applications. Yet, there is a growing interest in developing means for achieving “acoustic cloaking” of an animated boundary. We suggest the heating of a surface, generating thermoacoustic perturbations, as a mechanism for monitoring vibroacoustic sound. Considering a setup of an infinite planar wall interacting with a semi-infinite expanse of an ideal gas, we investigate the system response to arbitrary (small-amplitude) vibro-thermal excitation of the confining wall. Analysis is based on continuum Navier-Stokes-Fourier and kinetic Boltzmann equations, and supported by stochastic direct simulation Monte Carlo calculations. Starting with a case of a sinusoidally excited boundary, a closed-form solution is derived in both continuum and collision-free limits. The results, found valid at a wide range ...

Sound generated by a wing with a flap interacting with an eddy

Acoustic signature of a rigid wing, equipped with a movable downstream flap and interacting with a line vortex, is studied in a two-dimensional low-Mach number flow. The flap is attached to the airfoil via a torsion spring, and the coupled fluid-structure interaction problem is analysed using thin-airfoil methodology and application of the Brown and Michael equation. It is found that incident vortex passage above the airfoil excites flap motion at the system natural frequency, amplified above all other frequencies contained in the forcing vortex. Far-field radiation is analysed using Powell-Howe analogy, yielding the leading order dipole-type signature of the system. It is shown that direct flap motion has a negligible effect on total sound radiation. The characteristic acoustic signature of the system is dominated by vortex sound, consisting of relatively strong leading and trailing edge interactions of the airfoil with the incident vortex, together with late-time wake sound resulting from induced flap m...