A Justification of the Averaging Method for a Problem of Convection in a Field of Rapidly Oscillating Forces and for Other Parabolic Equations

Simonenko, I. B.

doi:10.1070/sm1972v016n02abeh001424

Cited by 53 publications

(27 citation statements)

References 3 publications

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“…The averaged equations are valid under the assumptions: (i) the frequency must be su ciently high (but not acoustic) that the vibration period is small compared to all the characteristic hydrodynamic times min(L 2 =a; L 2 =D; L 2 = ), (ii) the displacement amplitude must be small in the sense that bmin(L=ÿ T (Â 1 − Â 2 ); L=ÿ C (c 1 − c 2 )). Simonenko (1972) proves the convergence of solutions of the averaged system to averaged solutions of the system (3) -(6). A numerical description of this convergence is given by Khallouf (1995), while Gershuni and Lyubimov (1998) show that the new terms act as a vibrational body force directed opposite to the kinetic energy gradient.…”

Section: Problem Description and Basic Equationsmentioning

confidence: 99%

Natural doubly diffusive convection with vibration

et al. 2001

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We present a numerical and analytical study of doubly di usive convection driven by horizontal thermal and solutal gradients in square and rectangular enclosures with no-slip walls subjected to high-frequency vibration. The two vertical walls of the enclosure are maintained at di erent but uniform temperatures and concentrations while the horizontal walls are assumed to be impermeable and insulating. The resulting system is described by time-averaged Boussinesq equations. These equations possess a doubly di usive quasi-equilibrium solution provided the thermal and solutal buoyancy forces are equal and opposite. This solution is linearly stable up to a critical value of the stability parameter independently of the strength and orientation of the vibration. The solutions in the neighborhood of the bifurcation point are described analytically as a function of the strength and orientation of the vibration, and the larger amplitude states are computed numerically using a spectral collocation method. For vertical oscillation increasing the vibration amplitude decreases the subcriticality of the solutions and may even reverse it; the opposite occurs with horizontal vibration.

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Section: Problem Description and Basic Equationsmentioning

confidence: 99%

Natural doubly diffusive convection with vibration

et al. 2001

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“…Sometimes, an a posteriori justification for these formal conclusions is accessible (e.g. [10], [11]). Using the same formal approach we discuss the stability of relatively steady flows.…”

Section: S104mentioning

confidence: 99%

High-Frequency Asymptotic for the Forced Periodical Flows of an Inviscid Fluid through a Given Domain

Morgulis

2005

J. math. fluid mech.

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In this paper, we employ the method of multiple scales to study high-frequency timeperiodic solutions for a boundary value problem for Euler equations in a domain with permeable boundary. Here the permeability means that an inflow and an outflow of fluid through some parts of the boundary are allowed. The boundary data given at the flow inlet and at the flow outlet are subjected to a high-frequency time-periodic modulation while the fluid is supposed to be free of mass forces. The relevant example is a flow through a finite duct or pipe affected by the modulation of the inflow and of the outflow of fluid. The keynote of our study is that the fluid passing through a domain is a non-conservative system even if the fluid is inviscid and incompressible. Indeed, the pumping and dissipation of energy occur when the fluid particles are entering the domain and, correspondingly, leaving it. In particular, this means that the flow should react on the time-periodic forcing like a non-conservative system (e.g. a viscous fluid). But since we deal with an inviscid fluid, the high-frequency modulation should produce short fast waves that propagate from the inlet downstream. They contribute into the leading term of asymptotic in spite of small amplitude they have in the case under consideration. Moreover, this contribution admits a simple explicit expression which makes the asymptotic and stability analysis for the time-periodic regimes readily accessible. Mathematics Subject Classification (2000). 76E30 (37N10, 76B99).

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“…) the application of the averaging method of Simonenko [2] only allows the mean flow and mean temperature to be solved (see [21]). …”

Section: The Averaged Flow Equationsmentioning

confidence: 99%

“…The description of the thermovibrational flows in the limiting case of high-frequency and small amplitude of vibrations may be effectively obtained in the frame of the averaging method, which leads to the system of equations for averaged fields of velocity, pressure and temperature. Simonenko et al [1,2], were the first to use this method and propose the time-averaged form of the Boussinesq equations. Subsequently, it was demonstrated that high-frequency vibrations are most relevant in modifying stability characteristics.…”

Section: Introductionmentioning

confidence: 99%

05/00906 Rayleigh Bénard convective instability of a fluid under high-frequency vibration

2005

Fuel and Energy Abstracts

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The generation of two-dimensional thermal convection induced simultaneously by gravity and high-frequency vibration in a bounded rectangular enclosure or in a layer is investigated theoretically and numerically. The horizontal walls of the container are maintained at constant temperatures while the vertical boundaries are thermally insulated, impermeable and adiabatic. General equations for the description of the time-averaged convective flow and, within this framework, the generalized Boussinesq approximation are formulated. These equations are solved using a spectral collocation method to study the influence of vibrations (angle and intensity). Hence, a theoretical study shows that mechanical quasi-equilibrium (i.e., state in which the averaged velocity is zero but the oscillatory component is in general non-zero) is impossible when the direction of vibration is not parallel to the temperature gradient. In the other case, it is proved that the mechanical equilibrium is linearly stable up to a critical value of the unique stability parameter, which depends on the vibrational field. In this paper, it is shown that high-frequency vertical oscillations can delay convective instabilities and, in this way, reduce the convective flow. The isotherms are oriented perpendicular to the axis of vibration. In the case where the direction of vibration is perpendicular to the temperature gradient, small values of the Grashof number, the stability parameter, induce the generation of an average convective flow. When the aspect ratio is large enough, the character of the bifurcation is practically the same as in the limiting case of an infinitely long layer.

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A Justification of the Averaging Method for a Problem of Convection in a Field of Rapidly Oscillating Forces and for Other Parabolic Equations

Cited by 53 publications

References 3 publications

Natural doubly diffusive convection with vibration

Natural doubly diffusive convection with vibration

High-Frequency Asymptotic for the Forced Periodical Flows of an Inviscid Fluid through a Given Domain

05/00906 Rayleigh Bénard convective instability of a fluid under high-frequency vibration

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