Numerical study of convection of a liquid heated from below

The equilibrium of a liquid heated from below is stable only for small values of the vertical temperature gradient. With increase of the temperature gradient a critical equilibrium situation occurs, as a result of which convection develops. If the liquid fills a closed cavity, then there is a discrete sequence of critical temperature gradients (Rayleigh numbers) for which the equilibrium loses stability with respect to small characteristic disturbances. This sequence of critical gradients and motions may be found from the solution of the linear problem of equilibrium stability relative to small disturbances. If the temperature gradient exceeds the lower critical value, then (for steady-state heating conditions) there is established in the liquid a steady convective motion of a definite amplitude which depends on the magnitude of the temperature gradient. Naturally, the amplitude of the steady convective motion cannot be determined from linear stability theory; to find this amplitude we must solve the problem of convection with heating from below in the nonlinear formulation. A nonlinear study of the steady motion of a liquid in a closed cavity with heating from below was made in [1]. In that study it was shown that for Rayleigh numbers R which are less than the lower critical value R_c steady-state motions of the liquid are not possible. With R>R_c a steady convection arises, whose amplitude near the threshold is small and proportional to (R–R_c)^1/2 (the so-called soft instability)-this is in complete agreement with the results of the phenom-enological theory of Landau [2, 3].Primarily, various versions of the method of expansion in powers of the amplitude [4–8] have been used, and, consequently, the results obtained in those studies are valid only for values of R which are close to R_c, i. e., near the convection threshold.It is apparent that the study of developed convective motion far from the threshold can be carried out only numerically, with the use of digital computers. In [9, 10] the numerical methods have been successfully used for the study of developed convection in an infinite plane horizontal liquid layer.The present paper undertakes the numerical study of plane convective motions of a liquid in a closed cavity of square section. The complete nonlinear system of convection equations is solved by the method of finite differences on a digital computer for various values of the Rayleigh number, the maximal value exceeding by a factor of 40 the minimal critical value R_c. The numerical solution permits following the development of the steady motion which arises with R>R_c in the course of increase of the Rayleigh number and permits study of the oscillatory motions which occur at some value of the parameter R. The heat transfer through the cavity is studied. The corresponding linear problem on equilibrium stability is solved approximately by the Galerkin method.     

Onset of convection due to heating from above and heat release at an interface

A specific non-Rayleigh mechanism of convective instability in systems with an interface, developing in the presence of heating from above (anticonvection), was identified in [1, 2]. The onset of this type of instability requires a sharp difference in the physical parameters of the fluids. The effect of heat release and heat absorption on the onset of convection in systems for which this instability mechanism is possible is examined. In the presence of surface heat sources the directions of the temperature gradients in the two media may be different. The interaction of Rayleigh and non-Rayleigh types of instability is investigated. It is shown that for the water-mercury system on a certain interval of the parameters the oscillatory mode of instability is the most dangerous.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 16–20, May–June, 1990.     

Contraction of a highly nonequilibrium current-bearing plasma

Contracted is the term applied to that inhomogeneous state of a plasma in which it withdraws from the enclosing walls and concentrates in a more or less thin layer through which a current passes. Contraction is the result of instability developed in the original homogeneous state and may be related to the existence of a volt-ampere characteristic segment with negative differential conductivity. This phenomenon is known in semiconductor physics, and various instability mechanisms leading to contraction have been studied [1], Well known in a low-temperature plasma is thermal contraction connected with superheating instability of the electron gas [2–4]. In the present study we will consider a highly nonequilibrium plasma in which contraction may develop as a result of disproportion in the number of electrons, i.e., contraction of a recombination-ionization character. We consider below the homogeneous state of a nonequilibrium weakly ionized plasma with charged-particle concentration n_e- 10¹¹-10¹³ cm^–3 (electron temperature T of the order of thousands of degrees, with gas cold). Disequilibrium is produced by the departure of radiation beyond the limits of the plasma volume. Such a state will be considered with respect to the instability noted, but not studied, in [5]. As a consequence of this instability the plasma may transform to an inhomogeneous (contracted) state, which is considered under conditions such that Joulean electron heating is compensated by losses due to elastic collisions with atoms of the gas. Charge diffusion plays the basic role in establishing the boundaries dividing the currentbearing region from that without current. More complex is the situation where radiation losses of energy are also significant and superheating, as well as ionization instability, is possible. This case is evaluated briefly at the close of the study.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 45–54, January–February, 1975.     

Mean square velocities of bénard convection and heat transfer

The approximate formula K a^–2R(N–1), where a is a constant near 9 and R and N are the Rayleigh and Nusselt numbers, was proposed in [1] for the dimensionless kinetic energy K of convection in a horizontal layer of liquid. It is shown in the present paper that this expression is exact in linear and weakly nonlinear convection theory when the velocity and temperature fields are represented analytically [2–4]. The valuea is found to be 8.76 when the upper and lower boundaries of the layer are solid walls. The results are given of numerical calculations of the kinetic energy of the convection and the heat transfer in a wide range of Rayleigh numbers (up to 44 000) and Prandtl numbers (0.025 P 15). Analysis of the results shows that a is in fact a weak function of both R and P. If this is also the case at large R, it indicates a certain breaking of scaling of the mean convection characteristics at sufficiently large values of the Rayleigh number. It also indicates why laboratory experiments give values of n in the dependence N Rⁿ which are generally slightly less than the theoretical value n = 1/3.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 11–18, September–October, 1981.We should like to thank N. F. Vel'tishchev for providing first data of the numerical experiments of [13–15].     

Parametric excitation of waves at an interface

Experiments on the parametric excitation of waves at a fluid interface show a strong disagreement with theoretical results [1–3], since the latter do not take into account the influence of the second medium. This proves to be especially important at low frequencies. Thus, for a water-air interface with an excitation frequency = 60 sec^–1 the contribution amounts to 10%,and with = 30 sec^–1, even 20%. In this paper the stability of the interface of two viscous, incompressible fluids of finite depth in a variable gravity field is considered. The problem is put in the linear form by making an expansion with respect to the small viscosity and is solved by taking the Laplace transform with respect to time. A second-order integrodifferential equation with periodic coefficients is obtained for the deviation of the interface from the equilibrium position; its solution is sought by the method of averaging [4]. It is shown that the presence of the second fluid significantly raises the threshold of instability.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 167–170, March–April, 1977.     

Connective instability of a two-layer system with thermally insulated boundaries

The investigation of the convective stability of mechanical equilibrium of two horizontal layers of immiscible fluids has revealed the characteristics of such systems [1–3]. In particular, it has been found that, as distinct from a homogeneous horizontal layer, under certain conditions two-layer systems experience convective instability when uniformly heated from above and, moreover, oscillatory instability when heated from below. In [1–3] the problem was solved for a system with isothermal outer boundaries. In this paper the stability of equilibrium of two-layer systems is investigated for thermally insulated outer boundaries. Special attention is given to the study of the long wave instability mode.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 22–28, March–April, 1986.The authors wish to thank O. V. Kustova for assisting with the computations.     

Oscillatory thermocapillary instability of a liquid layer in the presence of a surfactant

The effect of capillarity and a surfactant on the stability of a liquid layer in the presence of a vertical temperature gradient is investigated. It is found that the surfactant leads to the appearance of both monotonic and oscillatory instability, the presence of a surface concentration destabilizing the equilibrium in the case of heating from below. When the free surface is heated, the surfactant stabilizes the capillary instability.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.1, pp. 6–10, January–February, 1993.     

Rheological properties of skim milk gels at various temperatures; interrelation between the dynamic moduli and the relaxation modulus

The rheological properties of rennet-induced skim milk gels were determined by two methods, i.e., via stress relaxation and dynamic tests. The stress relaxation modulusG _c (t) was calculated from the dynamic moduliG andG by using a simple approximation formula and by means of a more complex procedure, via calculation of the relaxation spectrum. Either calculation method gave the same results forG _c (t). The magnitude of the relaxation modulus obtained from the stress relaxation experiments was 10% to 20% lower than that calculated from the dynamic tests.Rennet-induced skim milk gels did not show an equilibrium modulus. An increase in temperature in the range from 20° to 35 °C resulted in lower moduli at a given time scale and faster relaxation. Dynamic measurements were also performed on acid-induced skim milk gels at various temperatures andG _c (t) was calculated. The moduli of the acid-induced gels were higher than those of the rennet-induced gels and a kind of permanent network seemed to exist, also at higher temperatures. G storage shear modulus,N·m^–2; - G loss shear modulus,N·m^–2; - G _c calculated storage shear modulus,N·m^–2; - G _c calculated loss shear modulus,N·m^–2; - G _e equilibrium shear modulus,N·m^–2; - G _ec calculated equilibrium shear modulus,N·m^–2; - G(t) relaxation shear modulus,N·m^–2; - G _c (t) calculated relaxation shear modulus,N·m^–2; - G ^*(t) pseudo relaxation shear modulus,N·m^–2; - H relaxation spectrum,N·m^–2; - t time,s; - relaxation time,s; - angular frequency, rad·s^–1. Partly presented at the Conference on Rheology of Food, Pharmaceutical and Biological Materials, Warwick, UK, September 13–15, 1989 [33].     

Unsteady wave motions of a fluid above an inclined floor

The propagation of unsteady waves above a flat inclined floor within the framework of a linear dispersion model was first studied in [1]. This paper shows how to solve the three-dimensional problem for the case = /4, where is the angle of inclination of the floor plane to the free surface. The two-dimensional problem was studied in [2–4]. In articles [2, 3] asymptotic solutions were found for the Cauchy-Poisson problem for certain values of . In [4], a method is proposed for solving the problem of the wave motion of a fluid due to the displacement of a section of the floor of the basin. However, the complicated structure of the expression obtained by reducing the problem to an inhomogeneous functional equation makes it impossible to study the solution. The aim of the present work is to obtain some exact solutions for the two- and three-dimensional problems of unsteady waves above an inclined floor, which are suitable for calculations and asymptotic estimates.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 65–70, November–December, 1984.     

Asymptotic solution to the problem of hypersonic flow over bodies in the neighborhood of the separation point of a thin shock layer

A study is made of the problem of hypersonic flow of an inviscid perfect gas over a convex body with continuously varying curvature. The solution is sought in the framework of the asymptotic theory of a strongly compressed gas [1–4] in the limit M when the specific heat ratio tends to 1. Under these assumptions, the disturbed flow is situated in a thin shock layer between the body and the shock wave. At the point where the pressure found by the Newton-Buseman formula vanishes there is separation of the flow and formation of a free layer next to the shock wave [1–4]. The singularity of the asymptotic expansions with respect to the parameter ₁ = ( –1)/( + 1) associated with separation of the strongly compressed layer has been investigated previously by various methods [3–9]. Local solutions to the problem valid in the neighborhood of the singularity have been obtained for some simple bodies [3–7]. Other solutions [7, 9] eliminate the singularity but do not give the transition solution entirely. In the present paper, an asymptotic solution describing the transition from the attached to the free layer is constructed for a fairly large class of flows.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 99–105, January–February, 1982.     

Flow of a viscous fluid in a closed cavity in the presence of slip effects

Slip at the wall is observed in the flow of non-Newtonian fluids [1–4] and rarefied gases [5]. The most complete information on the phenomenon is obtained in capillary viscosimetry. For small radii of the capillaries and in porous media the slip effect is manifested even for Newtonian fluids (water, kerosene, for example) [6]. Experiments [2, 4] show that the influence of the entrance section can be ignored if the length of the capillary exceeds its radius by about 100 times. For the measurement of the rheological characteristics of high-viscosity fluids the use of long capillaries is difficult, and it is necessary to calculate the two-dimensional flow at the entrance section with allowance for slip. The need for such calculations also arises, for example, when one is choosing the optimal parameters of the screw devices employed in the processing of polymers [7]. Two-dimensional flows of a viscous incompressible fluid are frequently calculated with the flow function and vorticity =– used as variables [8–14]. The expressions for the vorticity on the boundary are usually obtained from the viscous no-slip condition [8, 9]. In the present paper, expressions are obtained for the vorticity on a wall in the presence of slip. The obtained expressions are used to solve a test problem on the flow of a viscous incompressible fluid in a cavity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 10–16, January–February, 1980.     

On the initial stage of a point explosion in a radiating gas

A study is made of the initial stage of a point explosion in a radiating gray gas whose absorption coefficient is approximated by the dependenceK=x()e ^–n ,where is the density and e is the internal energy of the gas. It is shown that for n > —1/3 the initial stage of the process differs significantly from the solution of the problem in not only the classical adiabatic case [1, 2] but also in the case of a medium with nonlinear thermal conductivity [2–4]. The supply of energy to the medium at a point leads to instantaneous heating of the complete medium. The form of this heating is found analytically. The method of matched asymptotic expansions is used to investigate the behavior of the solution in the neighborhood of the center. It is found that for definite conditions at the center of the perturbed region there are formed a shock wave and a region of reverse flow of the gas.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 75–82, May–June, 1980.I should like to thank V. P. Korobeinikov for interest in the work and a helpful discussion of it.     

Interaction of a shock front with a wavy contact discontinuity

The excitation of Richtmayer-Meshkov instability fora ₀^–1>1 (a ₀ are the amplitude and wavelength of the initial contact discontinuity) is experimentally and numerically investigated. The effect of the curvature of the initial contact discontinuity on the development of the instability is determined.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.5, pp. 168–174, September–October, 1992.     

Formation of a large radiation-induced fire

In connection with ecological and climatic problems and, in particular, the nuclear winter theory much attention has recently been paid to the study of the possible consequences of large fires and explosions [1–5]. Below, on the basis of the model proposed in [6, 7] we investigate the initial stage of development of a fire induced by the light emitted by a powerful explosion in the atmosphere. The situation above the seat of combustion (the aerodynamics of the layer of air adjacent to the ground, the configuration and characteristics of the smoke cloud formed) is analyzed.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 17–25, January–February, 1992.     

Propagation of small disturbances in a relaxing and radiating diatomic gas in vibrational nonequilibrium

In radiation gasdynamical problems, where the primary object of investigation is a moving gas, the influence of radiation on the parameters of the gas flow is usually neglected to avoid overcomplication of the problem. The growth and behavior of initial disturbances in a scattering, radiating, absorbing, viscous, heat-conducting gas characterized by local thermodynamic equilibrium has been investigated previously [1]. However, for low pressures (p10^–4 to 10^–3 technical atm) and fairly high temperatures of the active molecular degrees of freedom (T10³ to 3·10³K) the distribution of the molecules among the vibrational levels can differ markedly from the equilibrium distribution due to the or der-of-magnitude closeness of the vibrational relaxation time _c associated with collisions and the radiative deactivation time * of excited molecules [2, 3]. We now analyze normal modes in a vibrationally nonequilibrium medium with allowance for radiation scattering in the vibrational-rotational band. We formulate a dispersion relation and discuss some limiting cases.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 168–171, September–October, 1976.The author is grateful to V. I. Kruglov, Yu. V. Khodyko, and M. A. El'yashevich for their interest and discussions.     

Laminar boundary layer on blunt bodies with account for vorticity of the external flow

This paper presents the technique for and results from numerical calculations of the hypersonic laminar boundary layer on blunted cones with account for the vorticity of the external flow caused by the curved bow shock wave. It is assumed that the air in the boundary layer is in the equilibrium dissociated state, but the Prandtl number is assumed constant, =0.72. The calculations were made in the range of velocities 3–8 km/sec, cone half-angles _k=0°–20°. With account for the vortical interaction of the boundary layer with the external flow, the distribution of the thermal flux and friction will depend on the freestream Reynolds number (other conditions being the same). In the calculations the Reynolds number R, calculated from the freestream parameters and the radius of the spherical blunting, varies from 2.5·10³ to 5.10⁴. For the smaller Reynolds numbers the boundary layer thickness on the blunting becomes comparable with the shock standoff, and for R<2.5·10³ it is apparent that we must reconsider the calculation scheme. With R>5·10⁴ for cones which are not very long the vortical interaction becomes relatively unimportant. The results of the calculations are processed in accordance with the similarity criteria for hypersonic viscous gas flow past slender blunted cones [1, 2].     

Existence and Uniqueness of Time-Periodic Physically Reasonable Navier-Stokes Flow Past a Body

Let be a three-dimensional exterior domain of class C^2,, 0<<1. Assume that a Navier-Stokes liquid is moving in under the action of a body force F that is time-periodic of period T, and that the velocity of the liquid is zero at spatial infinity. In this paper we show that, if F satisfies suitable conditions, and its norm, in appropriate function spaces, is sufficiently small, there is at least one time-periodic strong solution. Furthermore, the velocity field v of such a solution decays to zero for large |x| as |x|^–1 and its spatial gradient decays as |x|^–2, both uniformly in time. In addition, the pressure p decays like |x|^–2 and its gradient like |x|^–3, for almost all t[0,T]. In the special case where F is time-independent, these solutions are also time-independent and coincide with Finns physically reasonable solutions [4]. Moreover, we show that our time-periodic solutions are unique in a very large class, namely, the class of time-periodic weak solutions satisfying the energy inequality and with corresponding pressure fields verifying mild summability conditions in ×[0,T].     

Numerical investigation of the hydrodynamic drag of a circular cylinder coated with a thin layer of magnetic fluid

In order to reduce the drag of bodies in a viscous flow it has been proposed to apply to the surface exposed to the flow a layer of magnetic fluid, which can be retained by means of a magnetic field and thus act as a lubricant between the external flow and the body [1, 2]. In [1] the hydrodynamic drag of a current-carrying cylindrical conductor coated with a uniform layer of magnetic fluid was theoretically investigated at small Reynolds numbers. In order to simplify the equations of motion, the Oseen approximation was introduced for the free stream and the Stokes approximation for the magnetic fluid [3]. This approach has led to the finding of an exact analytic solution from which it follows that at Reynolds numbers Re 1 the drag of the cylinder can be considerably reduced if the viscosity of its magnetic-fluid coating is much less than the viscosity of the flow. The main purpose of the present study is to investigate, with reference to the same problem, how the magnetic-fluid coating affects the hydrodynamic drag at Reynolds numbers 1 Re 10²–10³, i.e., under separated flow conditions. In this case the simplifications associated with neglecting the nonlinear inertial terms in the Navier—Stokes equation are inadmissible, so that a solution can be obtained only by numerical methods.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 11–16, May–June, 1986.     

Convection in a horizontal fluid layer with specific-volume inversion

The effect of the position of the inversion point within the layer on the critical values of the Rayleigh number and the amplitudes of the rectangular-cell convective flows is numerically investigated. The monotonic instability of the mechanical equilibrium of the fluid with respect to small perturbations periodic along the layer is studied by the linearization method. The Lyapunov-Schmidt method is used to construct the secondary steady convective flows. The applicability of these methods in incompressible fluid stability problems was demonstrated in [8–10]. The calculations show that, starting from a certain value of the parameter , the branching is subcritical for any cell side ratio and a fixed wave vector modulus. For smaller values of the nature of the branching depends on the cell side ratio. This points to subcritical branching and hysteresis effects in those cases in which the periodicity of the perturbations is determined by external factors (corrugation of the boundary, spatially periodic temperature modulation, etc.). It is noted that the rectangular convection amplitude tends to zero when the cell side ratio tends to 3, the value at which hexagonal cellular convection is possible.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 43–49, January–February, 1989.The author wishes to thank V. I. Yudovich for his interest and useful advice and the participants in the Rostov State University Computational Mathematics Department's Scientific Seminar for discussing the results.     

Supersonic air flow past a sphere with account for vibrational relaxation

A numerical solution is obtained for the problem of air flow past a sphere under conditions when nonequilibrium excitation of the vibrational degrees of freedom of the molecular components takes place in the shock layer. The problem is solved using the method of [1]. In calculating the relaxation rates account was taken of two processes: 1) transition of the molecular translational energy into vibrational energy during collision; 2) exchange of vibrational energy between the air components. Expressions for the relaxation rates were computed in [2]. The solution indicates that in the state far from equilibrium a relaxation layer is formed near the sphere surface. A comparison is made of the calculated values of the shock standoff with the experimental data of [3].Notation uV_max, vV_max velocity components normal and tangential to the sphere surface - V_max maximal velocity - P V _max ² pressure - density - TT temperature - e_viRT vibrational energy of the i-th component per mole (i=–O₂, N₂) - =rb–1 shock wave shape - a _f the frozen speed of sound - HRT/m gas total enthalpy