Nonlinear interaction of longitudinal—Transverse acoustic waves in a circular cylinder

The limiting amplitudes of acoustic oscillations in a cylindrical volume of a heat releasing medium in which one or several modes are unstable in the linear approximation are determined. One of the mechanisms limiting the amplitudes of unstable acoustic modes is the transfer of energy from them to damped modes by nonlinear interaction. The nonlinear interactions of plane acoustic waves in a long channel have been considered by Artamonov and Vorob'ev [1]; in the present paper, the interaction of mixed longitudinal—transverse acoustic modes in a closed cylindrical volume is considered. The equations describing the interaction of two and three longitudinal—transverse modes are derived and investigated in the quadratic approximation by the method of slowly varying amplitudes and phases of the oscillations [2]. The treatment is applicable to a high-temperature gas, for which general stability conditions in the linear approximation have been formulated by Artamonov [3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 3–9, September–October, 1982.I should like to express my thanks to K. I. Artamonov (deceased) for suggesting the problem and for scientific supervision and A. P. Vorob'ev for constant interest in the work and helpful advice.     

Energy consideration of the nonlinear effects in a Rijke tube

The goal of this work is to characterize the excited states of a thermoacoustic system with mean flow. The properties of excited regimes are determined by the balance between thermoacoustic energy transformation and acoustic losses. In many systems, the sound intensity is not sufficient for nonlinear acoustic losses to be a major factor in defining nonlinear saturation of thermoacoustic instability. It is the nonlinearity of the heat transfer process that is responsible for limit-cycle stabilization of linearly unstable acoustic modes and for the appearance of higher harmonics. In the present study, both a nonlinear theory based on energy consideration and a model for the nonlinear convective heat transfer in unsteady flow are developed. Experimental data are obtained for the excited regimes of operation of an electric Rijke tube. Model results for hysteresis in the transition between stable and excited states and for limit-cycle parameters are compared with test data.     

Nonsteady burning of a solid propellant in a gaseous oxidizer flow

The burning of a solid propellant is investigated for nonsteady heat propagation in the induction zone. The equation of heat conduction in the propellant is solved in finite form for the case of a sharp change in burning rate; the time dependence of the temperature gradient at the propellant surface is obtained and used to investigate the mechanism of collapse of the diffusion flame above the surface. The combustion stability of a propellant burning in a channel with a large free volume is analyzed. The perturbations of the gas-dynamic quantities are related with the perturbations of the burning rate and hence with the properties of the induction zone in the solid phase. An analysis of the dispersion relation for the limiting case of propagation of acoustic waves in a stationary gas shows that the longitudinal acoustic perturbations that develop in the channel may grow with time, interacting with the heated subsurface layer of propellant.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fizikt, No. 4, pp. 44–52, July–August, 1971.In conclusion the author thanks B. V. Librovich for formulating and discussing the problem and A. G. Istratov and V. G. Markov for their valuable comments.     

Forced Convection Heat Transfer of Nanofluids in a Porous Channel

This article is concerned with the effects of flow and migration of nanoparticles on heat transfer in a straight channel occupied with a porous medium. Investigation of force convective heat transfer of nanofluids in a porous channel has not been considered completely in the literature and this challenge is generally considered to be an open research topic that may require more study. The fully developed flow and steady Darcy?CBrinkman?CForchheimer equation is employed in porous channel. The thermal equilibrium model is assumed between nanofluid and solid phases. It is assumed that the nanoparticles are distributed non-uniformly inside the channel. As a result the volume fraction distribution equation is also coupled with governing equations. The effects of parameters such as Lewis number, Schmidt number, Brownian diffusion, and thermophoresis on the heat transfer are completely studied. The results show that the local Nusselt number is decreased when the Lewis number is increased. It is observed that as the Schmidt number is increased, the wall temperature gradient is decreased and as a consequence the local Nusselt number is decreased. The effects of Lewis number, Schmidt number, and modified diffusivity ratio on the volume fraction distribution are also studied and discussed.     

Laminar flow and heat transfer in confined channel flow past square bars arranged side by side

A numerical investigation was conducted to analyze the unsteady laminar flow field and heat transfer characteristics in a plane channel with two square bars mounted side by side to the approaching flow. A finite volume technique is applied with a fine grid and time resolution. The transverse separation distance between the bars (G/d) is varied from 0 to 5, whereas the bar height to channel height is d/H=1/8, and the channel length is L=5H. Different flow regimes develop in the channel due the interaction between the two mounted square bars, steady flow, flow with vortex shedding synchronization either in phase or in anti-phase, or biased flow with low frequency modulation of vortex shedding are found. Results show that the pressure drop increase and heat transfer enhancement are strongly dependent of the transverse separation distance of the bars and the channel Reynolds number.     

Forced Convection with Slip-flow in a Channel or Duct Occupied by a Hyper-porous Medium Saturated by a Rarefied Gas

An analytic solution is obtained for forced convection flow in a parallel-plates channel or a circular duct occupied by a hyper-porous medium saturated with a rarefied gas in the slip-flow regime, for the case of uniform flux boundary conditions. As expected, it is found that velocity slip leads in general to increased heat transfer and temperature slip leads to reduced heat transfer.     

Numerical Study of Heat Transfer in a Rectangular Channel with Porous Covering Obstacles

A numerical study is performed to analyze steady laminar forced convection in a channel in which discrete heat sources covered with porous material are placed on the bottom wall. Hydrodynamic and heat transfer results are reported. The flow in the porous medium is modeled using the Darcy–Brinkman–Forchheimer model. A computer program based on control volume method with appropriate averaging for diffusion coefficient is developed to solve the coupling between solid, fluid, and porous region. The effects of parameters such as Reynolds number, Prandtl number, inertia coefficient, and thermal conductivity ratio are considered. The results reveal that the porous cover with high thermal conductivity enhances the heat transfer from the solid blocks significantly and decreases the maximum temperature on the heated solid blocks. The mean Nusselt number increases with increase of Reynolds number and Prandtl number, and decrease of inertia coefficient. The pressure drop along the channel increases rapidly with the increase of Reynolds number.     

Global modes in a confined impinging jet: application to heat transfer and control

We investigate the stability and control of a plane, laminar jet impinging on a flat plate in a channel, a geometry used to cool down a hot wall with a cold air jet in many industrial configurations. The global stability analysis indicates that, even for a strong confinement, the two-dimensional (2-D) steady flow is unstable to three-dimensional (3-D), steady perturbations. In the simplest limit case where dilatation effects are neglected, we show that the development of the instability induces a significant spanwise modulation of the heat flux at the impacted wall. To control the leading global mode, we propose adjoint-based 3-D harmonic and 2-D steady forcing in the bulk or at the wall. We show for instance that the unstable mode is controllable using a spanwise uniform blowing at the upper wall, in a specific domain corresponding to the footprint of the upper recirculating bubble. These techniques are applied to a novel open-loop control, in which we introduce into the flow a small airfoil, modelled by the lift force it exerts on the flow.     

Dependence of the natural frequencies and mode shapes of vibrations of an ideal gas on the acoustic resonance parameters

The problem of plane wave propagation through a circular hole is studied in the framework of long-wave approximation. The constructive notion of “apparent mass of holes” (Rayleigh; Fok) is used to construct a mathematical model of gas vibrations in an acoustic resonator and determine and analyze the natural frequencies and mode shapes for the velocity potential depending on the relative geometric parameters of the system. The high-precision calculations of the boundary value problem for the natural frequencies and mode shapes in the parametric approximation to the cross-section are based on a numerical-analytical accelerated convergence method. Two models are analyzed and compared, and the basic qualitative properties of gas vibrations are revealed depending on the basic parameters such as the mode number, relative size of the hole, and the dividing wall location.     

Unsteady waves in a liquid containing vapor bubbles

Unsteady wave processes in vapor-liquid media containing bubbles are investigated taking into account the unsteady interphase heat and mass transfer. A single velocity model of the medium with two pressures is used for this, which takes into account the radial inertia of the liquid with a change in volume of the medium and the temperature distribution in it [1]. The system of original differential equations of the model is converted into a form suitable for carrying out numerical integration. The basic principles governing the evolution of unsteady waves are studied. The determining influence of the interphase heat and mass transfer on the wave behavior is demonstrated. It is found that the time and distance at which the waves reach a steady configuration in a vapor-liquid bubble medium are considerably less than the correponding characteristics in a gas-liquid medium. The results of the calculation are compared with experimental data. The propagation of acoustic disturbances in a liquid with vapor bubbles was studied theoretically in [2]. The evolution of waves of small but finite amplitude propagating in one direction in a bubbling vapor-liquid medium is investigated in [3, 4] on the basis of the generalization of the Burgers-Korteweg-de Vries equation obtained by the authors. An experimental investigation of shock waves in such a medium is reported in [5, 6], and the structure of steady shock waves is discussed [7].Translated from Izvestiya Akademii Nauk SSSR, Hekhanika Zhidkosti i Gaza, No. 5, pp. 117–125, September–October, 1984.     

Laminar mixed convection on a horizontal plate of finite length in a channel of finite width

The steady two-dimensional laminar mixed-convection flow past a horizontal plate of finite length is analysed for large Péclet numbers, small Prandtl numbers and weak buoyancy effects. The plate is placed in a channel of finite width, with the plane walls of the channel being parallel to the plate. The temperature of the plate is assumed to be constant. The hydrostatic pressure difference across the wake behind the plate is compensated by a perturbation of the inviscid channel flow. This outer flow perturbation affects the temperature distribution in the thermal boundary layer at the plate and the heat transfer rate, respectively. Solutions in closed form are given. The forces acting on the plate due to the potential flow perturbation are also determined.     

Three-wave interactions of disturbances in a hypersonic boundary layer on a porous surface

Interactions of disturbances in a hypersonic boundary layer on a porous surface are considered within the framework of the weakly nonlinear stability theory. Acoustic and vortex waves in resonant three-wave systems are found to interact in the weak redistribution mode, which leads to weak decay of the acoustic component and weak amplification of the vortex component. Three-dimensional vortex waves are demonstrated to interact more intensively than two-dimensional waves. The feature responsible for attenuation of nonlinearity is the presence of a porous coating on the surface, which absorbs acoustic disturbances and amplifies vortex disturbances at high Mach numbers. Vanishing of the pumping wave, which corresponds to a plane acoustic wave on a solid surface, is found to assist in increasing the length of the regions of linear growth of disturbances and the laminar flow regime. In this case, the low-frequency spectrum of vortex modes can be filled owing to nonlinear processes that occur in vortex triplets.     

Analysis of a self-excited system with dry friction

A new approach to the modelling of systems in flow-induced self-excitation is presented. An elastically supported body situated in a channel carrying a flowing medium is analysed and the effect of additional dry friction is investigated. It is shown that in some flow velocity intervals several steady solutions of the mathematical model (locally stable equilibrium positions and steady vibrations) can exist. Their domains of attraction are determined.     

On the flow of a viscous fluid driven along a channel by suction at porous walls

This paper is a theoretical treatment of the flow of a viscous incompressible fluid driven along a channel by steady uniform suction through porous parallel rigid walls. Many authors have found such flows when they are symmetric, steady and two-dimensional, by assuming a similarity form of solution due to Berman in order to reduce the Navier-Stokes equations to a nonlinear ordinary differential equation. We generalise their work by considering asymmetric flows, unsteady flows and three-dimensional perturbations. By use of numerical calculations, matched asymptotic expansions for large values of the Reynolds number, and the theory of dynamical systems, we find many more exact solutions of the Navier-Stokes equations, examine their stability, and interpret them. In particular, we show that most previously found steady solutions are unstable to antisymmetric two-dimensional disturbances. This leads to a pitchfork bifurcation, stable asymmetric steady solutions, a Hopf bifurcation, stable time-periodic solutions, stable quasi-periodic solutions, phase locking and chaos in succession as the Reynolds number increases.     

Approximate analytical calculation of the mach configuration of steady shock waves in a plane constricting channel

An approximate analytical model for calculation of the parameters of a steady gas flow inside a plane constricting channel formed by two symmetrically positioned wedges is suggested. A Mach configuration of shock waves (triple point) is formed in the channel when the wedge angles are larger than some critical value. The flow calculation in a constricting channel reduces to the solution of the iterative problem for a system of nonlinear algebraic equations. The configurations of shock waves, the slipstream, and the sonic line are described by the proposed model of a gas flow. A comparison of the results obtained using this model allows a fairly accurate calculation of the Mach stem and the length of the subsonic-flow region. Institute of Theoretical and Applied Mechanics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 3, pp. 52–58, May–June, 1998.     

Finite Element Analysis of Convection Flow Through a Porous Medium in a Horizontal Channel

We analyse the convection flow of a viscous fluid through a horizontal channel enclosing a fully saturated porous medium. The Galerkin finite element analysis is used to discuss the flow and heat transfer through the porous medium using serendipity elements. The velocity, the temperature distributions and the rate of heat transfer are analysed for variations in the governing parameters. The profiles at different vertical levels are asymmetric curves, exhibiting reversal flow everywhere except on the midplane. In a given porous medium, for fixed G or N, the temperature in the fluid region at any position in fluids with a higher Prandtl number, is much higher than in fluids with a lower Prandtl number. Likewise, other parameters being fixed, lesser the permeability of the medium, lower the temperature in the flow field. Nu reduces across the flow at all axial positions, while it enhances along the axial direction of the channel. Nu reduces with decrease in the Darcy parameter D, and thus lesser the permeability of the medium, lesser the rate of heat transfer across the boundary at any axial position of the channel.     

Chaos in Acoustic Subspace Raised by the Sommerfeld–Kononenko Effect

This paper deals with vibrations of an infinite plate in contact with an acoustic medium where the plate is subjected to a point excitation by an electric motor of limited power-supply. The whole system is divided into two “exciter - foundation” and “foundation-plate-medium”. In the system “motor-foundation” three classes of steady state regimes are determined: stationary, periodic and chaotic. The vibrations of the plate and the pressure in the acoustic fluid are described for each of these regimes of excitation. For the first class they are periodic functions of time, for the second they are modulated periodic functions, in general with an infinite number of carrying frequencies, the difference between which is constant. For the last class they correspond to chaotic functions. In another mathematical model where the exciter stands directly on an infinite plate (without foundation) it was shown that chaos might occur in the system due to the feedback influence of waves in the infinite hydro-elastic subsystem in the regime of motor shaft rotation. In this case the process of rotation can be approximately described as a solution of the fourth order nonlinear differential equation and may have the same three classes of steady state regimes as the first model. That is the electric motor may generate periodic acoustic waves, modulated waves with an infinite number of frequencies or chaotic acoustic waves in a fluid.     

Coupled free vibration analysis of a fluid-filled rectangular container with a sagged bottom membrane

In the present paper, two-dimensional coupled free vibrations of a fluid-filled rectangular container with a sagged bottom membrane are investigated. This system consists of two rigid walls and a membrane anchored along two rigid vertical walls. It is filled with incompressible and inviscid fluid. The membrane material is assumed to act like an inextensible material with no bending resistance. First, the nonlinear equilibrium equation is solved and the equilibrium shape of the membrane is obtained using an analytical formulation neglecting the membrane weight. The small vibrations about the equilibrium configuration are then investigated. Along the contact surface between the bottom membrane and the fluid, the compatibility requirement is applied for the fluid–structure interactions and the finite element method is used to calculate the natural frequencies and mode shapes of the fluid–membrane system. The vibration analysis of the coupled system is accomplished by using the displacement finite element for the membrane and the pressure fluid-finite element for the fluid domain. The variations of natural frequencies with the pressure head, the membrane length, the membrane weight and the distance between two rigid walls are examined. Moreover, the mode shapes of system are investigated.     

Heat transfer in a gas in the presence of acoustic turbulence

An equation for the average internal energy of a gas in a field of acoustic turbulence is obtained by the method of perturbation theory. It is shown that, in addition to the characteristic increase in the coefficient of thermal conductivity, acoustic turbulence leads to heating of the gas through compressibility and heat-conduction effects. At large Mach and Péelet numbers the heating has an exponential character with time. An expression determining the absorption of acoustic vibrations in a gas is obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gasa, No. 1, pp. 183–187, January–February, 1978.     

Effect of static loading on the nonlinear vibrations of a three-time redundant portal frame: Analytical and numerical studies

An analytical study of the nonlinear vibrations in a three-time redundant portal frame is presented herewith, considering the effect of the axial forces caused by the static loading upon the first anti-symmetrical mode (sway) and the first symmetrical mode natural frequencies. It is seen that the axial forces may play an important role in tuning the sway mode and the first symmetrical mode into a 1:2 internal resonance. Harmonic support excitations resonant with the first symmetrical mode are then introduced and the amplitudes of nonlinear steady states are computed based upon a multiple scales solution. Comparisons with numerical analyses using a finite-element program developed by the authors show good qualitative agreement.