Instability of small-amplitude convective flows in a rotating layer with free boundaries

The stability of steady convective flows in a horizontal layer with free boundaries, heated from below and rotating about a vertical axis, is studied in the Boussinesq approximation (Rayleigh-Bénard convection). The flows considered are convective rolls or square cells that are sums of two perpendicular rolls with equal wave numbers k. It is assumed that the Rayleigh number is almost critical in order for convective flows with a wave number k: R = R _c (k) + ε² to arise, the amplitude of the supercritical states being of the order of ε. It is shown that the flows are always unstable relative to perturbations that are the sum of one long-and two short-wave modes corresponding to linear rolls turned through small angles in opposite directions.     

Convection in rotating spherical layers with allowance for centrifugal forces

The investigation of convection in rotating spherical layers with a central gravitational field g(r) is very important for the study of the global motions in the atmospheres of large planets and the convective zones of stars. In recent years, many studies of these questions have been made (they have been reviewed, for example, by Yavorskaya and Belyaev [1]), but the centrifugal convective force has been ignored in all the numerical and analytic investigations. In some cases, for example, for large planets, the centrifugal force may reach an appreciable value, O.1g, and have a strong influence on the convective motion. The present paper studies the occurrence of convection in slowly rotating spherical layers with allowance for centrifugal forces. It is shown that the centrifugal force leads to the appearance in a layer of an axisymmetric flow, at the stability limit of which convective cells of banana or toroidal shape can develop. The latter are possible only in layers with undeformable boundaries at sufficiently large values of the Froude number. Irrespective of the form of the layer and the magnitude of the centrifugal force, the banana-shaped cells propagate in a wavelike manner in the direction opposite to the rotation. In the case of undeformable boundaries, the centrifugal force stabilizes the motion of the fluid as compared with the case of a layer at rest. Deformation of one or both of the boundaries under the influence of the centrifugal force leads to destabilization of the basic flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 14–21, March–April, 1984.     

Axisymmetric convection in a spherical layer: Transitions between steady-state regimes

The steady-state convective motions of a viscous fluid occupying a spherical layer R₁ r R₂, R₂/R₁=1.2 are studied. The non-deformable boundaries of the layer are assumed to be free of shear stresses. At the outer boundary the constant temperature and at the inner boundary the constant heat flux are given. The system of equations in the Boussinesq approximation is solved by the Galerkin method with time stabilization on the assumption of axial and equatorial symmetry. It is shown that at the point Ra=Ra_c the state of mechanical equilibrium loses stability and steady symmetrical supercritical bifurcation is observed. The modes most unstable in the linear sense determine the form of convection when Ra > Ra_c and the supercriticality is not too great. At Rayleigh numbers Ra_c < Ra < 200Ra_c there exists a set of steady-state solutions with different spatial structures. The realization of solutions of a particular type depends on the supercriticality and the initial conditions. The evolution of the solutions with variation of the Rayleigh number is investigated. The changes in the spatial kinetic energy spectra and the integral heat fluxes upon transition from one branch of the solutions to another and with variation of the supercriticality are analyzed. As the supercriticality increases, despite the excitation of more and more new small-scale modes, the large-scale motions begin to make an ever greater contribution to the total energy. The results obtained can be used for constructing hydrodynamic models of the global motions in the atmospheres of giant planets, the convective envelopes of stars, and in the depths of the earth's mantle.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 17–24, November–December, 1989.     

Stability of wavy conditions in a film flowing down an inclined plane

The nonlinear theory of motion in a film of liquid flowing down an inclined plane predicts the existence of an interval k₀m, inside of which the wave number of periodic wave motion may lie [1]. The condition of the stability of experimentally attained motions imposes a limitation on their wave numbers. In [2] a numerical investigation of the stability of wavy motions was made; in the investigated range of change in the Galileo number and the wave number all the motions were found to be unstable; however, the fastest growing were perturbations imposed on a motion with a determined wave number (“optimal” conditions). In [3] the instability of motions with a wavelength exceeding some limiting value was established in a long-wave approximation. In the present work, within the framework of the two-dimensional problem, an investigation was made of the stability of periodic wavy motions, based on expansion in terms of the small parameter k_m. It is established that, within the interval k₀m, there lies a finite subinterval of wave numbers for which wavy motions are stable. The narrowness of this interval (δk≈0.07 k_m) may be the reason why, in the experiment, with not too great Galileo numbers for fully established periodic wavy motions, no substantial differences in the wave-length are observed [4].     

Theory of gravitational stability of a rotating cylinder

The stability of a rotating dust cylinder against perturbations located in the plane perpendicular to the axis of rotation is investigated. It is shown that a homogeneous rotating cylinder containing a weak inhomogeneity is stable against such perturbations. A weakly inhomogeneous cylinder with opposite streams of equal density is unstable for thel=2 mode in the case of a perturbation of the form_ei(l–t), when the density increases radially. The instability of a system consisting of a homogeneous rotating dust cylinder in a hot homogeneous medium is determined. It is shown that the maximum growth rate corresponds tol = 2 when the density of a cold cylinder is not negligible in comparison with the density of the medium. In the opposite case, the maximum growth rate shifts toward l=3. An attempt is made to associate the existence of the maximum growth rate for l=2 with the presence of two spiral arms in most galaxies. It is shown that, when the longitudinal temperature is high enough, a rotating cylinder which is bounded in the radial direction is stable against arbitrary perturbations.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol.10, No. 3, pp. 3–11, May–June, 1969.     

Effect of Low Rotation Rate on Steady Convection During the Solidification of a Ternary Alloy

We consider the problem of steady convective flow during the directional solidification of a horizontal ternary alloy system rotating at a constant and low rate about a vertical axis. Under the limit of large far-field temperature, the flow region is modeled to be composed of two horizontal mushy layers, which are referred to here as a primary layer over a secondary layer. We first determine the basic state solution and then carry out linear stability analysis to calculate the neutral stability boundary and the critical conditions at the onset of motion. We find, in particular, that there are two flow solutions and each solution exhibits two neutral stability boundaries, and the flow can be multi-modal in the low rotating rate case with local minima on each neutral boundary. The critical Rayleigh number and the wave number as well as the vertical volume flux increase with the rotation rate, but the flow is found to be less stabilizing as compared to the binary alloy counterpart flow. The effects of low rotation rate increase the solid fraction and the liquid fraction at certain vertically oriented fluid lines, and the highest value of such increase is at a horizontal level close to the interface between the two mushy layers.     

Investigation on the mechanism of convective mass transfer from a horizontal rotating cylinder without air jet flow

The local and mean convective mass transfer coefficients from the surface of a large-diameter horizontal circular rotating cylinder without air jet flow were investigated by measuring the concentration gradient. The results indicate that rotation performs different effects on the convective mass transfer at different regions. Based on the experimental data, the correlation equations of the mean convective mass transfer Sherwood number Sh and the critical Reynolds number Re _r,cri have been formulated as follows: $ Sh = 0.32[(8.5Re_{\text{r}}^{2} + Gr) \cdot Sc]^{1/3} $ and $ Re_{\text{r,cri}} = 0.44(Gr \cdot Sc)^{1/2} $ .     

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.     

Instability of the motion of a liquid between two rotating spherical surfaces

A study has been made of the stability of the steady-state motion of a viscous incompressible liquid, arising in a thin spherical layer, when both spheres are rotating in the same direction at different angular velocities. For a ratio of the radii of the spheres r₂/r₁=1.10, 1.07, a stability curve is obtained which is analogous to the stability curve for the motion of a liquid between rotating cylinders.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 155–156, July–August, 1970.     

The effect of vertical vibrations on the onset of convection in a planar rotating layer

The effect of vertical vibrations on the convection in a rotating planar fluid layer heated from below was studied. In this case a modulation parameter, the acceleration due to gravity, appears in the problem. The modulation of the parameter may have a significant effect on the onset of convective instability. Parameter modulation in nonrotating layers has been investigated in earlier work [1–3]. The presence of rotation significantly increases the complexity of the mathematical problem, introducing an additional dependence of the solution on the Taylor number Ta and the Prandtl number Pr. Furthermore, an oscillatory convection regime can occur at the stability limit in rotating fluids with Pr < 1. Parameter modulation in the rotating fluid may not only lead to a change in the stability limit and critical wavelength but also to a change in the eigenfrequency of the oscillatory convection. Rauscher and Kelly [4] examined the effect of parameter modulation on the convective stability of a rotating fluid only for the particular case of a sinusoidal variation in the temperature gradient with a small amplitude for Pr = 1, i.e., the effect of modulation was studied on only a steady convection regime.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 12–22, July–August, 1984.     

On the nature of the boundary layers instabilities in a flow between a rotating and a stationary discSur la nature des instabilités de couche limite dans un écoulement entre un disque fixe et un disque tournant

Both theoretical linear stability analysis and direct numerical simulation are performed to study the transition flow between a stationary and a rotating disc. This paper concerns three-dimensional spiral and annular patterns computed with a high-order (spectral) numerical method and related to Bödewadt layer instabilities. The characteristic parameters of these boundary layer patterns are compared to the theoretical results and interpreted in terms of type I and type II generic instabilities. Moreover, the absolute instability regions are also theoretically identified and the critical Reynolds numbers of the convective/absolute transition in both layers are given. To cite this article: E. Tuliska-Sznitko et al., C. R. Mecanique 330 (2002) 91–99.     

Axisymmetric thermal stress in a thin spherical shell by the method of matched asymptotic expansions

A solution of the equations of linear Thermoelasticity is presented for a closed shell with constant material properties. The solution is constructed by matching asymptotic expansions in the thinness parameter (h/a = thickness/radius of curvature) in the various regions of the shell. For clamped conditions at the (meridional opening angle-constant) edge (θ = θ₀), the solution has the character expected of a thin shell, i.e. a membrane region in the interior with a “thin shell” boundary layer near θ = θ₀. For the stress-free condition, however, an “Elasticity” layer of meridional width of order h must be introduced between the “thin-shell” layer and the edge (θ = θ₀). This solution is also compared with an asymptotic solution of the thin-shell equations and shown to agree through two orders of magnitude of h/a^1/2.     

Temperature correlations with vorticity and velocity in a turbulent cylinder wake

This work aims to understand the difference in the correlations between the fluctuating temperature and the vorticity from that between the fluctuating temperature and the velocity in a turbulent cylinder near wake. Measurements are made at x/d = 10, 20 and 40, where x is the streamwise distance from the cylinder axis and d is the cylinder diameter, with a Reynolds number of 2.5×10³ based on d and the free-stream velocity. The three components of the fluctuating velocity vector u_i(i = 1, 2 and 3), vorticity vector ω_i (i = 1, 2 and 3), and temperature θ in the plane of the mean shear are measured simultaneously with a multi-wire probe consisting of four X-hotwires and four cold wires. It is found that at x/d = 10, both correlations between u_iand θ and between ω_i and θ predominantly take place at St = 0.21, due to the concentric distribution of the Kármán vortices and the heat. With increasing x/d, the correlation between ω_i (i = 1, 2 and 3) and θ drops rapidly, as a result of the weakened Kármán vortices; in contrast, the correlation between u₁ and θ increases appreciably, largely due to an enhanced correlation between u₁ and θ at low frequencies or scales of motions larger than the Kármán vortex. The slowly decreasing (along x) two-point autocorrelations of u₁ and θ suggest that the very-large-scale motions (VLSMs) found in wall flows occur also in the turbulent wake and are responsible for the high correlation between u₁ and θ at low frequencies.     

Stability of a fluid in a horizontal saturated porous layer: effect of non-linear concentration profile, initial, and boundary conditions

Carbon dioxide injected into saline aquifers dissolves in the resident brines increasing their density, which might lead to convective mixing. Understanding the factors that drive convection in aquifers is important for assessing geological CO₂ storage sites. A hydrodynamic stability analysis is performed for non-linear, transient concentration fields in a saturated, homogenous, porous medium under various boundary conditions. The onset of convection is predicted using linear stability analysis based on the amplification of the initial perturbations. The difficulty with such stability analysis is the choice of the initial conditions used to define the imposed perturbations. We use different noises to find the fastest growing noise as initial conditions for the stability analysis. The stability equations are solved using a Galerkin technique. The resulting coupled ordinary differential equations are integrated numerically using a fourth-order Runge–Kutta method. The upper and lower bounds of convection instabilities are obtained. We find that at high Rayleigh numbers, based on the fastest growing noise for all boundary conditions, both the instability time and the initial wavelength of the convective instabilities are independent of the porous layer thickness. The current analysis provides approximations that help in screening suitable candidates for homogenous geological CO₂ sequestration sites.     

A linear stability analysis on the onset of thermal convection of a fluid with strongly temperature-dependent viscosity in a spherical shell

A linear stability analysis was performed in order to study the onset of thermal convection in the presence of a strong viscosity variation, with a special emphasis on the condition for the stagnant-lid (ST) convection where a convection takes place only in a sublayer beneath a highly viscous lid of cold fluid. We consider the temporal evolution (growth or decay) of an infinitesimal perturbation superimposed to a Boussinesq fluid with an infinite Prandtl number which is in a static (motionless) and conductive state in a basally heated planar layer or spherical shell. The viscosity of the fluid is assumed to be exponentially dependent on temperature. The linearized equations for conservations of mass, momentum, and internal (thermal) energy are numerically solved for the critical Rayleigh number, Ra _c , as well as the radial profiles of eigenfunctions for infinitesimal perturbations. The above calculations are repeatedly carried out by systematically varying (i) the magnitude of the temperature dependence of viscosity, E, and (ii) the ratio of the inner and outer radii of the spherical shell, γ. A careful analysis of the vertical structure of incipient flows demonstrated that the dominance of the ST convection can be quantitatively identified by the vertical profile of Δ _h (a measure of conversion between horizontal and vertical flows), regardless of the model geometries. We also found that, in the spherical shell relevant to the Earth’s mantle (γ = 0.55), the transition into ST convection takes place at the viscosity contrast across the layer ${r_\eta\simeq10^4}$ . Taken together with the fact that the threshold value of r _η falls in the range of r _η for a so-called sluggish-lid convection, our finding suggests that the ST-mode of convection with horizontally elongated convection cells is likely to arise in the Earth’s mantle solely from the temperature-dependent viscosity.     

Turbulent wall pressure reduction using suction control

A study of the fluctuating wall pressure beneath a 2-d turbulent boundary layer was conducted in a water tunnel with Reynolds numbers, based on momentum thickness, ranging between 2,100 and 4,300. The boundary layer was perturbed with steady mild suction to assess the effect of upstream suction on the fluctuating wall pressure measured downstream of the suction slit. Wall pressure signatures were captured using a custom-fabricated piezo-ceramic array with d ⁺ values ranging between 64 and 107. Likewise, the velocity field was measured with a laser Doppler velocimeter with l ⁺ values ranging between 4.0 and 6.7 for the lowest and highest Re _θ investigated. Estimates of the wall pressure spectra revealed a noticeable hydrodynamic peak that scaled reasonably well with outer variables and with an average convective speed of 75 % of the free stream velocity (based on unconditionally sampled pressure time series). Two boundary layer suction control cases were studied corresponding to suction rates of less then 30 % of the boundary layer momentum. The findings reveal how only modest amounts of suction are needed to reduce upwards 50–60 % of the hydrodynamic ridge.     

Critical heat flux in saturated forced convective boiling on a heated disk with an impinging jet

The present study introduces a new correlation predicting critical heat flux (CHF) for a saturated forced convective boiling with an impinging jet. The new correlation is able to predict all the CHF data inV-regime with a good accuracy of +-20% to which the correlations existing until now could not be applicable for 15l/?_r<100 and D/d>36. The new correlation seems to support a new criterion of CHF mechanism applicable for not only pool boiling but also forced convective boiling, recently proposed by Katto and Haramura.     

LES of turbulent flow in a concentric annulus with rotating outer wall

Fully-developed turbulent flow in a concentric annulus, r₁/r₂ = 0.5, Re_h = 12,500, with the outer wall rotating at a range of rotation rates N = U_θ,wall/U_b from 0.5 up to 4 is studied by large-eddy simulations. The focus is on the effects of moderate to very high rotation rates on the mean flow, turbulence statistics and eddy structure. For N up to ∼2, an increase in the rotation rate dampens progressively the turbulence near the rotating outer wall, while affecting only mildly the inner-wall region. At higher rotation rates this trend is reversed: for N = 2.8 close to the inner wall turbulence is dramatically reduced while the outer wall region remains turbulent with discernible helical vortices as the dominant turbulent structure. The turbulence parameters and eddy structures differ significantly for N = 2 and 2.8. This switch is attributed to the centrifuged turbulence (generated near the inner wall) prevailing over the axial inertial force as well as over the counteracting laminarizing effects of the rotating outer wall. At still higher rotation, N = 4, the flow gets laminarized but with distinct spiralling vortices akin to the Taylor–Couette rolls found between the two counter-rotating cylinders without axial flow, which is the limiting case when N approaches to infinity. The ratio of the centrifugal to axial inertial forces, Ta/Re² ∝ N² (where Ta is the Taylor number) is considered as a possible criterion for defining the conditions for the above regime change.     

Convective stability of a horizontal rotating fluid layer with distributed heat sources

The results of investigating the convective instability of a horizontal layer of rotating fluid, created by a temperature difference applied at the boundaries of the layer and by heat sources distributed according to various laws, are presented. It is shown that, when the other parameters of the problem are fixed, an increase in the internal heat release lowers the limits of both monotonic and oscillatory stability of the layer, increases the wave number and reduces the neutral oscillation frequency. An increase in source concentration towards the center of the layer intensifies the effect. As the strength of the internal heat sources and their concentration towards the center of the layer increase, the oscillating convection that develops at the stability limit when the Prandtl number is low and the rotation fairly fast is first replaced by monotonic convection and then ceases altogether.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 21–28, January–February, 1989.