Elastic stress analysis of non-linear variable thickness rotating disks subjected to thermal load and having variable density along the radius

An analytical procedure for evaluation of elastic stresses and strains in non-linear variable thickness rotating disks, either solid or annular, subjected to thermal load, and having a fictitious density variation along the radius is presented. Thickness variation of disks is described by means of a power of linear function, which can be used to describe a fourfold infinity of actual disk profiles. The procedure is based on two independent integrals of the hypergeometric differential equation describing the displacement field; this theoretical procedure is just general and does not present limitations and drawbacks of the approaches as the one found in technical literature. General unpublished relations of stress state and displacement field in non-linear variable thickness disks subjected, under elastic conditions, to thermal gradient, and featuring a density variation along the radius are defined. Particular consideration is given to some industrial example of turbine rotors carrying hub and rim with buckets on periphery or radial blades on lateral surfaces. The analytical results obtained by using the new general relations perfectly match those obtained by FEA and overlap those concerning the special cases of tapered conical disks found in literature.     

Elastic stress analysis of rotating converging conical disks subjected to thermal load and having variable density along the radius

An analytical procedure for evaluation of elastic stresses and strains in rotating conical disks, either solid or annular, subjected to thermal load, and having a fictitious density variation along the radius is presented. The procedure is based on two independent integrals of the hypergeometric differential equation describing the displacement field; this procedure is just as general as the one found in technical literature, but less intricate and more reliable. General unpublished relations of stress state and displacement field in conical disk subjected, under elastic conditions, to thermal gradient, and featuring a density variation along the radius are defined. Particular consideration is given to some industrial example of turbine rotors carrying hub and rim with buckets on periphery or radial blades on lateral surfaces. The analytical results obtained by using the new general relations perfectly match those obtained by FEA.     

Convection and heat transfer in a vertical layer with allowance for radiation from nonisothermic walls

Many studies (for example, [1–5]) consider motion and heat transfer in closed vertical cavities with given different temperatures of the lateral boundaries. The majority of studies cover the case of convection, but of late studies have appeared (for example, [4]) in which joint radiative—convective heat transfer is taken into account. In the present study we consider motion and heat transfer in a rectangular cavity separating two media with given different temperatures. In contrast to [4], the temperature of the lateral boundaries is determined from the condition for interaction with the surrounding medium, and the air in the cavity is assumed to be transparent for the heat radiation of the walls. The problem considered is a mathematical model of the heat transfer through windows, and is necessary for the analysis of methods of improving the heat proofing of buildings.Translated-from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 25–30, 1987.     

Convective stability of a nonisothermal fluid in a rotating horizontal coaxial gap

The thermal fluid convection in a coaxial horizontal gap uniformly rotating about its axis is investigated. The threshold above which convective flows are excited and the structure of these flows are studied. It is found that convection ensues irrespective of whether the inner or outer boundary temperature is higher. Convection manifests itself in the threshold development of rolls elongated in the direction of the rotation axis and is determined by two different mechanisms. If the layer is heated from outside, the centrifugal convection mechanism plays a leading part and the diameter of the convective rolls is comparable with the layer thickness. If the higher is the temperature of the inner boundary of the layer, the centrifugal inertia force has a stabilizing effect and convection development is related with the action of thermal vibrational mechanism. The latter is determined by gravity-generated oscillations of the nonisothermal fluid relative to the cavity. The wave number of the vibrational convective structures is several times smaller than under centrifugal convection. The results obtained broaden our understanding of thermal convection in systems rotating in external static force fields.     

Convective heat transfer in a rotating horizontal cylindrical fluid layer

This paper describes the thermal convection and heat transfer in a cylindrical fluid layer rotating around a horizontal axis, with various constant temperatures set at the layer boundaries. The influence of the rotational speed of the cylindrical fluid layer on the convective heat transfer in this layer is studied. The study results are presented as functions of dimensionless parameters that characterize the action of two convective mechanisms: centrifugal and thermal-oscillatory. It is shown that, with low rotational speed, the heat transfer is determined by quasistationary gravitational convection.     

Numerical investigation of convective motion in a closed cavity

The investigation of thermal convection in a closed cavity is of considerable interest in connection with the problem of heat transfer. The problem may be solved comparatively simply in the case of small characteristic temperature difference with heating from the side, when equilibrium is not possible and when slow movement is initiated for an arbitrarily small horizontal temperature gradient. In this case the motion may be studied using the small parameter method, based on expanding the velocity, temperature, and pressure in series in powers of the Grashof number—the dimensionless parameter which characterizes the intensity of the convection [1–4]. In the problems considered it has been possible to find only two or three terms of these series. The solutions obtained in this approximation describe only weak nonlinear effects and the region of their applicability is limited, naturally, to small values of the Grashof number (no larger than 10³).With increase of the temperature difference the nature of the motion gradually changes—at the boundaries of the cavity a convective boundary layer is formed, in which the primary temperature and velocity gradients are concentrated; the remaining portion of the liquid forms the flow core. On the basis of an analysis of the equations of motion for the plane case, Batchelor [4] suggested that the core is isothermal and rotates with constant and uniform vorticity. The value of the vorticity in the core must be determined as the eigenvalue of the problem of a closed boundary layer. A closed convective boundary layer in a horizontal cylinder and in a plane vertical stratum was considered in [5, 6] using the Batchelor scheme. The boundary layer parameters and the vorticity in the core were determined with the aid of an integral method. An attempt to solve the boundary layer equations analytically for a horizontal cylinder using the Oseen linearization method was made in [7].However, the results of experiments in which a study was made of the structure of the convective motion of various liquids and gases in closed cavities of different shapes [8–13] definitely contradict the Batchelor hypothesis. The measurements show that the core is not isothermal; on the contrary, there is a constant vertical temperature gradient directed upward in the core. Further, the core is practically motionless. In the core there are found retrograde motions with velocities much smaller than the velocities in the boundary layer.The use of numerical methods may be of assistance in clarifying the laws governing the convective motion in a closed cavity with large temperature differences. In [14] the two-dimensional problem of steady air convection in a square cavity was solved by expansion in orthogonal polynomials. The author was able to progress in the calculation only to a value of the Grashof numberG=10⁴. At these values of the Grashof numberG the formation of the boundary layer and the core has really only started, therefore the author's conclusion on the agreement of the numerical results with the Batchelor hypothesis is not justified. In addition, the bifurcation of the central isotherm (Fig. 3 of [14]), on the basis of which the conclusion was drawn concerning the formation of the isothermal core, is apparently the result of a misunderstanding, since an isotherm of this form obviously contradicts the symmetry of the solution.In [5] the method of finite differences is used to obtain the solution of the problem of strong convection of a gas in a horizontal cylinder whose lateral sides have different temperatures. According to the results of the calculation and in accordance with the experimental data [9], in the cavity there is a practically stationary core. However, since the authors started from the convection equations in the boundary layer approximation they did not obtain any detailed information on the core structure, in particular on the distribution of the temperature in the core.In the following we present the results of a finite difference solution of the complete nonlinear problem of plane convective motion in a square cavity. The vertical boundaries of the cavity are held at constant temperatures; the temperature varies linearly on the horizontal boundaries. The velocity and temperature distributions are obtained for values of the Grashof number in the range 0<G4·10⁵ and for a value of the Prandtl number P=1. The results of the calculation permit following the formation of the closed boundary layer and the very slowly moving core with a constant vertical temperature gradient. The heat flux through the cavity is found as a function of the Grashof number.     

Convection experiments in an inclined narrow cavity

The liquid flow behaviour in a small vertical gap with a heated and a cooled sidewall was studied experimentally in a former work as far as heat and mass transfer are concerned [Heiland et al. in Heat Mass Transf 43:863–870, 2007]. Following this, the study of thermal convection in a narrow cavity with variable inclination angle has been performed with liquid crystal techniques. Velocity and temperature fields of the flow have been measured. The results show that the strongest convection intensity arises in a vertical cavity.     

Asymptotic analysis of convective flows in a rotating spheroid for vanishing viscosity and thermal conductivity

The six-dimensional model of viscous fluid thermal convection in a uniformly rotating ellipsoid [1] is studied. The limiting case in which the viscosity and thermal conductivity approach zero while the Prandtl number Pr remains finite is considered. Using both asymptotic and numerical methods, it is shown that for Pr > 2 the attractor is a two-dimensional invariant torus or a limit cycle; the corresponding convective flows are either quasiperiodic, with two basic frequencies, or periodic. It is also shown that resonances on the torus play a dominant role in the breakdown of two-dimensional toruses with increasing viscosity and thermal conductivity.Rostov-on-Don. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 35–38, September–October, 1995.     

Vibrational thermal convection in a rectangular cavity

Vibrational thermal convection in a rectangular cavity under conditions of weightlessness is studied. Some equilibrium configurations were obtained in earlier papers of two of the authors [1, 2] and their linear stability investigated. In the present paper, a numerical investigation is made of the developed vibrational convection which arises under conditions when equilibrium is impossible. The structure of the average vibrational-convective flows and the characteristics of the heat transfer are determined. The change of regimes and the connection with the stability problem are discussed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 94–99, July–August, 1982.     

Dynamical responses of non-isothermal flow between two independently rotating disks with time-dependent wall conditions

The present work develops a theoretical model of rotational convection and uses it to investigate the dynamical responses of the flow and heat transfer between two disks rotating at different rates under the influences of time-dependent disturbances. The unsteady non-isothermal flow model is formulated by extending a recently developed steady-state similarity model of axi-symmetric rotational convection. In the new model all the rotation-induced buoyancy forces are considered. Using one disk as reference, effects of the time-dependent changes in wall temperature or rotating rate of the other disk on the flow and heat transfer are explored. Various rotational modes with asymptotic or fluctuating change in boundary condition of temperature or disk rotation are studied. The present time-dependent model for this non-isothermal rotating flow is numerically solved by a finite-difference method. By using the present results, the complicated flow and heat transfer mechanisms with thermal-flow coupling in the class of time-dependent rotational convection are manifested.     

Numerical analysis of the stability of nonisothermal couette flow

The stability of convective motion of a liquid between two rotating heated cylinders is investigated in the absence of external forces. The mathematical model for describing the convection is obtained from the general equations [1, 2] on the assumption that the density of the liquid, the thermal conductivity, the specific heat and the viscosity coefficients depend only on temperature, and that the work done by the pressure forces and the viscous dissipation are negligibly small. The thermal expansion coefficient of the liquid is not assumed to be small, which distinguishes the models in question from the classical Oberbeck-Boussinesq model [1, 3, 4]. Rostov-on-Don. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 70–76, September–October, 1988.     

Thermally Induced,Nonlinear Vibrations of Rotating Disks

The natural frequency and responses for the nonlinear free vibration ofheated rotating disks are presented analytically when nonuniformtemperature distributions pertaining to the laminar and turbulentairflow induced by disk rotation are considered. The nonuniformtemperature distributions on the disk are highly dependent on itsrotation speed. The natural frequencies for symmetric and asymmetricresponses of a 3.5 inch diameter computer memory disk are calculated.When the disk is heated, its stiffness becomes larger for the two lowestnodal diameter numbers and smaller for the other nodal diameter numbers.It implies that the vibration of heated, rotating disks for the highernodal diameter numbers may be induced more easily than the cooled one.The results for the nonlinear vibration can reduce to those for thelinear vibration when the nonlinear effects vanish. To furtherinvestigate of the interaction of thermal and nonlinearity of rotatingdisks, the temperature distribution for such a rotating disk needs to bedeveloped.     

Bifurcation regimes of free convection in the presence of thermal radiation and cavity inclination

The bifurcation regimes of free convection in closed cavities with heating from below have been investigated numerically by many authors [1]. In the situations considered the equilibrium solution conditions were disturbed by only one factor, e.g. the inclination of the cavity to the vertical, the motion of one of the boundaries, a change in the equilibrium temperature distribution, etc. In this paper, the simultaneous influence of two factors that disturb the fluid equilibrium conditions, namely thermal radiation and a slight inclination of the cavity relative to the vertical, are investigated. It is shown that, for the simultaneous action of two destabilizing factors, a near-equilibrium solution is possible. Perm’. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 42–47, January–February, 2000. The work received financial support from the Russian Foundation for Basic Research (project No. 96-01-01737).     

Motion of a fluid between rotating cones

A study is made of the steady axisymmetric flow of a viscous fluid between two cones rotating in opposite ways round a common axis. It is shown that as in the case of the flow of fluid swirled by plane disks rotating at different speeds [1], there can be two regimes of motion in the system: a Batchelor regime with quasirigid rotation of the fluid outside the boundary layers [2] and a Stewartson regime in which the azimuthal flow is concentrated only in the boundary layers [3]. In the Stewartson regime, a boundary layer analogous to that in the single disk problem (see, for example, [4–6]) forms in the region of each cone far from the apex. For the flows outside the boundary layers, simple expressions are found which make it possible to obtain a conception of the circulation of the fluid as a whole. With minor alterations, the results can be applied to the case of the rotation of other curved surfaces.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 58–64, March–April, 1985.The author thanks A. M. Obukhov for suggesting the subject and for his interest in the work, and A. V. Danilov and S. V. Nesterov for useful discussions.     

Threshold excitation of photoabsorption convection

The article considers questions of the stability of the equilibrium states of a liquid which absorbs light. Threshold values are found for the intensity of the light in the problem of the stability of the equilibrium of a liquid in a square cavity with three thermally insulated walls. A steady-state integro-interpolation scheme is presented for the numerical calculation of problems of photoabsorption convection. The propagation of light waves in absorbing media is accompanied by the dissipation of radiant energy. In heavy liquids, absorption heating of a substance in the field of a wave may be the reason for the appearance of convection [1–3]. It is important to study the conditions for the appearance and the special characteristics of this type of convection, and its inverse effect on the structure of the light field. The first problem is important when the light beams are regarded only as a source of convection [4], and the second in questions of the directed propagation of light [5] and of self-focusing phenomena [2, 3, 6–10]. For high-energy heat fluxes and a liquid with a strong temperature dependence of its dielectric permeability, the convective self-stress will be very considerable; in this case, both problems are mutually interconnected. The excitation of convection by the absorption of light, without taking account of the inverse effect on the structure of the light beam, was studied numerically in [1, 4]. Equations for photoabsorption convection, taking account of convective self-stress in the Boussinesq approximation and of the geometry of the optics, were formulated in [11]. Several economical finite-difference schemes for solving problems of photoabsorption convection problems in rectangular cavities are discussed in [12]. The present article is devoted to an investigation of the threshold intensities of light for the excitation of photoabsorption convection. The existence of critical intensities of light, above which the mechanically equilibrium states of the liquids absorbing the light become unstable, was demonstrated in [1, 4].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 128–135, September–October, 1971.The authors thank A. V. Lykov for his continuing interest and aid, and G. I. Petrov and V. I. Polezhaev for their useful evaluation of the work.     

Convection in a closed cavity heated from below when equilibrium conditions are disrupted

The equilibrium of a fluid is possible in a closed cavity in the presence of a strictly vertical temperature gradient (heating from below) [1]. There is a distinct sequence of critical Rayleigh numbers R_i at which this equilibrium loses its stability relative to low characteristic perturbations. The presence of different finite perturbations, unavoidable in an experiment, leads to the absence of a strict equilibrium when R < R₁. The problem of the influence of the perturbation on the convection conditions near the critical points arises in this context [2, 3]. The case in which the cavity is heated not strictly from below is investigated in [2] and the case in which the perturbation of the equilibrium is due to the slow movement of the upper boundary of the region is investigated in [3]. In [2, 3] the perturbation has the structure of a first critical motion and thus the results of these papers coincide qualitatively. The perturbation of the temperature in the horizontal sections of the boundary, which creates a perturbation with a two-vortex structure corresponding to the second critical point R₂, is examined in this paper. A similar type of perturbation is characteristic for experiments in which the thermal conductivity properties of the fluid and the cavity walls are different. The nonlinear convection conditions are investigated numerically by the net-point method.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 203–207, March–April, 1977.The author wishes to thank D. B. Lyubimova, V. I. Chernatynskii, and A. A, Nepomnyashchii for their helpful comments.     

Laminar natural convection in a partially divided rectangular cavity at high Rayleigh number

Finite element predictions of two-dimensional laminar natural convection in a partially divided rectangular cavity at high Rayleigh number are presented. The walls are differentially heated, the horizontal surfaces are insulated and the cavity contains a partial vertical divider which is centrally located and whose height is varied. Detailed results are presented for an aluminium half-divider in water for Rayleigh number up to 10¹¹ and compared directly with recent experiments in a cavity of aspect ratio 1/2. The predicted flow and heat transfer are in good agreement with the measurements and confirm the existence of a high Rayleigh number regime with characteristic behaviour that differs significantly from that found at lower Rayleigh number. In addition, the effects of the divider height, the divider conductivity, the fluid Prandtl number and the cavity aspect ratio are studied. The results show that a direct simulation of the complex flow and heat transfer that occurs in partially divided cavities is possible for realistic physical conditions.     

Thermal structure of convective vortex lattices

The thermal structure of the convective motions of a rotating plane layer of fluid is experimentally investigated in the regular vortex structure regime. It is found that in such a system the intense vortex motion leads to a temperature distribution such that the mean fluid temperature falls linearly from the bottom of the layer to the surface, the temperature gradient being determined by the rate of rotation and depth of the fluid. By dimensional analysis it is shown that this gradient corresponds to heat transfer in which the Nusselt number isolines are parallel to the convection curve. The horizontal structure of the temperature field is investigated; it corresponds to motion in which the fluid descends within a narrow vortex-sink and rises along the edges of a cylinder which determines the characteristic dimension of the structure in rotating fluid convection.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 160–166, November–December, 1987.The author wishes to thank G. S. Golitsyn for his constant interest in the work.     

Buoyancy Convection in Low Prandtl Number Liquids with Large Temperature Variation

Two-dimensional laminar convection in low Prandtl number liquids driven by the buoyancy force is studied. The liquid is contained in a closed square cavity with isothermal vertical walls kept at different temperatures. The top and bottom walls are assumed to be insulated. The thermal conductivity of the liquid is assumed to depend on temperature. ADI and SOR schemes are employed. The heat transfer is found to decrease appreciably across the cavity with a decrease in thermal conductivity.