A non-reflective spectral wave maker for SPH modeling of nonlinear wave motion

A momentum source function derived from the linear wave theory is introduced into the weakly compressible smoothed particle hydrodynamic (SPH) model for the long-time simulation of regular and irregular wave generation problems. Wave absorption is realized by adding a velocity attenuation term into the governing momentum equation. The performances of the wave maker are tested under different wave conditions. The wave maker is then applied to the study of the challenging processes, such as random wave breaking on the reef-face or the reef-crest, wave setup and spectral transfer of wave energy from the peak frequency to lower frequencies over the reef-flat. The predicted results are compared with the experimental data and a good agreement is obtained. The SPH model with a non-reflective spectral wave maker can be used as a practical tool for studying wave interaction with coastal structures.     

Effect of capillary wave radiation on the oscillations of a vibrator

The oscillations of a rigid body on an elastic tie (vibrator) in an ideal incompressible fluid with a free boundary, on which surface tension forces act, are considered. The linearized problem of hydrodynamics is solved approximately in the self-consistent formulation, the reaction forces exerted on the body by the fluid are calculated, and an integrodifferential equation of motion is obtained. Using asymptotic methods, the average characteristics determining the damping coefficient and the frequency shift of the oscillations of the vibrator are obtained with allowance for the effect of the capillary waves radiated by the vibrator. Qualitative effects depending on the parameters of the system are revealed. The authors' numerical simulation of the motion of the vibrator completely confirms the qualitative conclusions concerning the nature of the oscillations of a body in a fluid having surface tension.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 126–132, March–April, 1995.     

Self-similar motion of a gas heated by a nonequilibrium continuous radiation spectrum

In this paper we discuss the motion of the vapor formed during the evaporation of a solid by a continuous radiation spectrum. The vapor is assumed to be heated by this radiation to a temperature T much higher than the phase-transition temperature T_v and much higher than the temperature T_i at which significant ionization of the vapor begins. in the case, T_v and T_i can be neglected (as can the heat of evaporation Q_v and the energy Q_i expended on ionization). As a result of this motion, the vapor has a density ? much lower than the densityρ ₀ of the solid. It can therefore be assumed that the heating wave moves through an absolutely cold and infinitely dense gas. At the same time, the vapor temperature is assumed low enough that reradiation can be neglected. The radiation-absorption coefficient η for the ionized vapor can be described by a power-law dependence on T and ? for certain ranges of T, ε, and the photon energy ε. In this case, the motion of the gas is a self-similar problem. The spectrum and angular distribution of the incident radiation [φ (ε, θ)] and the η and ε dependences can be arbitrary. A system of ordinary differential equations is found and solved. Intense radiation incident on a solid surface will evaporate the solid. If the absorption coefficient η of the vapor and the flux density q of the radiation are high enough, the escaping vapor will be heated to a high temperature in a relatively short time. This temperature will not only be much higher than the evaporation temperature T_v, but it will also be higher than the “ionization temperature” T_i. If the internal energy per unit mass of the vapor is much higher than the heat of evaporation Q_v and the energy Q_i expended on ionization, and if the vapor density ? is much lower than the initial densityρ ₀ as a result of its escape, then the problem of the motion and heating of the vapor can be simplified through the assumptions. (0.1) $$T_v = T_i = Q_v = Q_i = 0,\rho _0 = \infty (v_0 = 1/\rho _0 = 0)$$ (here and below, v is the specific volume). We can therefore assume that the heating wave moves through an infinitely dense and absolutely cold gas. In the region of multiple and complete ionization, the ionized-vapor absorption coefficient η, associated with free-free electron transitions in the field of ions, and bound-free transitions from the higherlying states of atoms and ions, has an approximately power-law dependence on T and ? [1], or on p and ? (p is the pressure): Here k and K are numerical coefficients which depend on the substance and on the ranges of T, ?, and ε in which (0.2) is used. For a completely ionized gas, we have α=3/2, β=1,a=?5/2, b=?3/2, and \( - \bar 1/2\) when ε?T; or α=3/2, β=1,a=3/2, and b=?1/2 when when ε ? T. We assume that (0.2) holds for any T, for approximation (0.1). We assume the ratio of specific heats γ to be constant for a certain temperature range in the range of multiple and complete ionization. With these simplifying approximations, the problem of the planar, transient flow of a gas heated by a beam of monochromatic radiation is a self-similar problem. It has been studied in [2,3]. It is shown below that the analogous problem of the motion of a gas heated by a nonequilibrium continuous radiation spectrum is also self-similar. For a partially ionized gas, approximation (0.2) is usualy satisfied only for the long-wavelength part of the incident spectrum. For the short-wavelength part of the spectrum (that is, for photons whose energy is close to or greater than the ionization potential characteristics of the ions for the given temperature range, and which are capable of direct photoionization of these ions from the ground or first excited status), the absorption coefficient is usually much smaller (by several orders of magnitude). This “hard” radiation penetrates a short distance into the solid, causing intense heating of a thin surface layer of small mass. An afterionization wave propagates through the substance, moving under the influence of the radiation flux in the hard part of the spectrum; if the temperature of the surface layer is close to the source temperature T_e, and reradiation becomes important, there will also be a thermal wave [1]. Since the energy expended in heating is large in these waves, their propagation velocity is small (in comparison with that of the wave of evaporation, initial ionization, and heating of the plasma by the long-wavelength part of the spectrum), even if the hard and soft parts of the incidence spectrum have comparable energies (E_h and E_s). Also, the intense reradiation by the thermal wave in the hard part of the spectrum increases its propagation velocity. Finally, the energy in the short-wavelength part of the spectrum may in general be small because of self-adsorption in the source itself (for example, adsorption of the short-wavelength radiation in the cold working gas ahed of a shock wave front in an explosive source [4]). Accordingly, the heating waves for the various parts of the source spectrum may propagate differently Since the mass of the surface layer heated by the short-wave-length part of the spectrum is small, the pressure produced as a result of of the disintegration of the surface layer is small when E_h is of the order order of E_s or, especially, when E_h?E_s; that is, the hydrodynamic effects of the heating and surface-layer disintegration on the motion and and heating of the deep layers heated by the “basic” part of the spectrum can also be neglecred. The high temperature and low density of this layer only facilitate the penetration of the long-wavelength part of the spectrum into the deeper layers; however, because of the small mass of this layer, even this phenomenon has little effect on the hydrodynamic processes in the deeper layers. Accordingly, Eq. (0.2) can frequently be assumed valid for the basic part of the spectrum in the case of a partially ionized gas, also; the rest of the spectrum may simply be neglected. These restrictions on the applicability of the self-similar problem are generally removed in the case of a completely ionized gas. A state close to that of complete ionization arises when two ionization potentials typical of a given temperature range are greatly different (this occurs, for example, in the case of the alkaline metals, and also when one atomic shell has been essentially ionized, while another has not yet started to be ionized; e. g., the L- and K-shells or the M- and L-shells). We consider here the case in which the heating is caused by nonequilibrium radiation, that is, radiation such that the intrinisic radiation of the vapor may be neglected. This is a valid assumption when the vapor temperature is considerably below the source temperature T_e, or, more accurately, when the following condition holds (for a Planckian source spectrum): (0.3) $$W\sigma T_e^4 \chi \left( {\frac{{\varepsilon _1 }}{{T_e }},\frac{{\varepsilon _2 }}{{T_e }}} \right) \gg \sigma \Upsilon ^4 \chi \left( {\frac{{\varepsilon _1 }}{T},\frac{{\varepsilon _2 }}{T}} \right)$$ Here W is the source-radiation dilution coefficient due to geometric factors, σ is the Stefan-Boltzmann constant, ?₁ and ?₂ are the boundaries of the “basic part” of the spectrum, and χ is the fraction of the spectral energy of a Planckian source with a temperature T_e or T for photous with energies ?₁≤?≤?₂. We note that the boundaries ?₁ and ?₂ for the source and vapor-radiation spectra are sometimes slightly differnt, but condition (0.3) can be easily modified for this situation or for a non-Planckian source spectrum. For our problem, the radiation intensity J=J (m, t, ε, θ) is a function of four variables: the time t, the Lagrangian mass coordinate m, the photon energy ε, and the angle θ between the direction of motion and the beam direction. The intensity J_o=J (o, t, ε, θ) of the radiation incident on the boundary m=0 is assumed to be a given function. In the self-similar problem, J can be represented as (0.4) $$J = t^\lambda J(mt^{ - n} ,\varepsilon ,\theta )$$ . This can be done (when conditions (0.1)–(0.3) are satisfied) when J_o can be represented by (0.5) $$J_0 = t^\lambda \psi (\varepsilon ,\theta )(\varepsilon _1 \leqslant \varepsilon \leqslant \varepsilon _2 ,\theta _1 \leqslant \theta \leqslant \theta _2 )$$ If the source spectrum is Planckian, condition (0.5) requires that T_e=const. In this case, the power-law time dependence of the intensity J_o may reflect, for example, motion of the radiation source toward the irradiated surface; in this case, however, the limiting angle θ₂ of the incident radiation also changes (usually, θ₁=0). As before, the problem is self-similar if these angles θ₂(t) are always small; that is, if the radiation is almost completely unidirectional. The arbitrary nature of the function ψ(ε, ч), which shows the spectrum and angular distribution of thesource radiation, and the arbitrary nature of the function φ (ε), which shows the dependence of the absorption coefficient on the photon energy, permit us to analyze the effects of these functions on the heating and motion of the substance for the case of the self-similar solution.     

Effect of surface gas condensation on the velocity of a reflected shock wave

The time-dependent one-dimensional problem of the normal reflection of a shock wave propagating at constant velocity in a gas (vapor) at rest from the plane surface of its condensed phase under steady-state condensation-evaporation conditions on the interphase plane is considered within the framework of the kinetic equation for a monatomic gas with a model collision operator (S-model). The solution is obtained using a conservative second-order finite-difference method. Attention is concentrated on the steady-state regime of the condensation process. The effect of the condensation (evaporation) coefficient on the velocity of the reflected shock wave is studied.     

Thermal study of an array of inline horizontal cylinders below a nearly adiabatic ceiling

Steady state two-dimensional free convection heat transfer from a horizontal, isothermal cylinder in a horizontal array of cylinders consists of three isothermal cylinders, located underneath a nearly adiabatic ceiling is studied experimentally. A Mach–Zehnder interferometer is used to determine thermal field and smoke test is made to visualize flow field. Effects of the cylinders spacing to its diameter (S/D), and cylinder distance from ceiling to its diameter (L/D) on heat transfer from the centered cylinder are investigated for Rayleigh numbers from 1500 to 6000. Experiments are performed for an inline array configuration of horizontal cylinders of diameters D = 13 mm. Results indicate that due to the nearly adiabatic ceiling and neighboring cylinders, thermal plume resulted from the centered cylinder separates from cylinder surface even for high L/D values and forming recirculation regions. By decreasing the space ratio S/D, the recirculation flow strength increases. Also, by decreasing S/D, boundary layers of neighboring cylinders combine and form a developing flow between cylinders. The strength of developing flow depends on the cylinders Rayleigh number and S/D ratio. Due to the developing flow between cylinders, the vortex flow on the top of the centered cylinder appears for all L/D ratios and this vortex influences the value of local Nusselt number distribution around the cylinder.Variation of average Nusselt number of the centered cylinder depends highly on L/D and the trend with S/D depends on the value of Rayleigh number.     

Numerical calculation of the propagation of a plane subsonic radiation wave through a gas in opposition to a flow of light radiation

The propagation of a plane heating and ionization wave through a gas is considered; the wave is sustained by a strong flow of monochromatic optical radiation (traveling in the opposite direction) through energy transfer attributable to the emission of a continuous spectrum. In the range of radiation flux densities under consideration, a situation arises in which the expanding hot layer generates a shock wave transparent to the incident radiation. The radiation wave is subsonic. The pressure within the hot layer is smoothly distributed, so that its parameters may be determined by considering the equations of energy and transport of the monochromatic source radiation and the radiative-transfer equations for various frequencies and directions. The true spectral composition and distribution of the radiation are considered in detail, using refined tables of the thermodynamic and optical properties. The results of numerical calculations relating to air are presented; so are certain details of the methods used in averaging the transfer equations, which prove very efficient for the radiation-gasdynamic problem under consideration and greatly reduce the volume of calculations.

     

Effect of external radiation on the distortion of a vitreous body in a flow of gas

The semitransparency of a material, which determines the penetration of external radiation into the inside layers, affects significantly the temperature profile in the body [1, 2]. Since the viscosity of a melt of viscous materials depends strongly on the temperature, deformation of the temperature profile close to the surface leads to considerable change of the rate of spread of the liquid film, which has a significant effect on the rate of distortion of the body. In the present paper, the problem of distortion is formulated taking into account the transfer of radiation inside the body. The dependence of the distortion parameters and the degree of blackness of the body on the fraction of radiation in the external thermal flux and the mean free path of the radiation inside the material is determined. A sufficient condition is also obtained for the presence of a temperature maximum inside the body in a more general case than in [3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 127–134, July–August, 1976.The author thanks G. A. Tirskii for a discussion of the posing of the problem.     

An investigation on near wall transport characteristics in an adiabatic upward gas–liquid two-phase slug flow

The near-wall transport characteristics, inclusive of mass transfer coefficient and wall shear stress, which have a great effect on gas–liquid two-phase flow induced internal corrosion of low alloy pipelines in vertical upward oil and gas mixing transport, have been both mechanistically and experimentally investigated in this paper. Based on the analyses on the hydrodynamic characteristics of an upward slug unit, the mass transfer in the near wall can be divided into four zones, Taylor bubble nose zone, falling liquid film zone, Taylor bubble wake zone and the remaining liquid slug zone; the wall shear stress can be divided into two zones, the positive wall shear stress zone associated with the falling liquid film and the negative wall shear stress zone associated with the liquid slug. Based on the conventional mass transfer and wall shear stress characteristics formulas of single phase liquid full-pipe turbulent flow, corrected normalized mass transfer coefficient formula and wall shear stress formula are proposed. The calculated results are in good agreement with the experimental data. The shear stress and the mass transfer coefficient in the near wall zone are increased with the increase of superficial gas velocity and decreased with the increase of superficial liquid velocity. The mass transfer coefficients in the falling liquid film zone and the wake zone of leading Taylor bubble are lager than those in the Taylor bubble nose zone and the remaining liquid slug zone, and the wall shear stress associated falling liquid film is larger than that associated the liquid slug. The mass transfer coefficient is within 10⁻³ m/s, and the wall shear stress below 10³ Pa. It can be concluded that the alternate wall shear stress due to upward gas–liquid slug flow is considered to be the major cause of the corrosion production film fatigue cracking.     

Free convection heat transfer from a horizontal fin attached cylinder between confined nearly adiabatic walls

Steady state two-dimensional free convection heat transfer from a horizontal, isothermal fin attached cylinder, located between nearly two adiabatic walls is studied experimentally using a Mach–Zehnder interferometer. Effects of the walls inclination angel (θ) on heat transfer from the cylinder is investigated for Rayleigh number ranging from 1000 to 15,500. Two cylinders with different diameters of D = 10 and 20 mm are used to cover wide Rayleigh range. Results indicate that, heat transfer phenomena differ for different Rayleigh number. For Rayleigh numbers lower than 5500, heat transfer rate from cylinder surface is lower than the heat transfer from a single cylinder. In this range by the use of walls, heat transfer from the cylinder decreases slightly and walls’ inclination does not change heat transfer rate from the cylinder surface. For Rayleigh number ranging from 5500 to 15,500, amount of heat transfer from the cylinder surface is less than that of a single cylinder. However, by adding nearly adiabatic walls to experimental model heat transfer mechanism differs and chimney effect between fin and walls increases the heat transfer rate from the cylinder surface. By increasing the walls inclination angel from 0° to 20°, the chimney effect between walls and fin diminishes and heat transfer rate from the cylinder surface is approaching to the heat transfer rate of fin attached cylinder without adiabatic walls.     

Group properties of the equations of adiabatic motion of a medium in relativistic hydrodynamics

A group classification is presented and the complete set of invariant solutions is found for the equations of adiabatic motion of a medium in relativistic hydrodynamics.     

Effect of tangential and normal forces on the wave formation in the flow of a liquid film and a gas

The effect of surface forces on nonlinear waves induced by the hydrodynamic instability in the flow of a viscous liquid film along the inner surface of a tube blown with a gas.     

Nonequilibrium radiation of a strong shock wave

We examine the nonequilibrium radiation from the first negative band of the molecular nitrogen ion N ₂ ⁺ . The various N ₂ ⁺ ion excitation mechanisms are discussed. It is shown that for a shock wave velocity in air 8 km/sec the primary excitation mechanism is electronic impact.In conclusion the authors wish to thank V. K. Vertushkin and A. A. Gladkov for supplying the calculations on the structure of the normal compression shock and L. I. Ponomarev for helpful discussions.