Stability of Thermal Vibrational Flow in an Inclined Liquid Layer Against Finite‐Frequency Vibrations

Stability of the flow that arises under the action of a gravity force and streamwise finitefrequency vibrations in a nonuniformly heated inclined liquid layer is studied. By the Floquet method, linearized convection equations in the Boussinesq approximation are analyzed. Stability of the flow against planar, spiral, and threedimensional perturbations is examined. It is shown that, at finite frequencies, there are parametricinstability regions induced by planar perturbations. Depending on their amplitude and frequency, vibrations may either stabilize the unstable ground state or destabilize the liquid flow. The stability boundary for spiral perturbations is independent of vibration amplitude and frequency.     

Natural convection in vertical slots with wall temperature oscillation

Natural convection in vertical slots has been studied experimentally with oscillation of a cold wall temperature. It is found that at sufficiently large Rayleigh numbers travelling-wave instability occurs, but only in the region close to the cold wall when the hot wall is maintained at the initial temperature, and appears in both cold and hot regions when both hot and cold wall temperatures are changed simultaneously and symmetrically with the initial temperature value, although the hot wall temperature is kept constant after the change. The observed instability may be attributed to the leading-edge effect induced by the cold wall temperature oscillation, and selectively amplified by the natural convection flow.The author would like to thank Professor R. S. Tankin for his encouragement and assistance throughout this study. The financial support of the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged.     

Penetrative convection in a horizontally isotropic porous layer

Penetrative convection in a horizontally isotropic porous layer is investigated primarily using an internal heat sink model and alternatively, a quadratic density temperature law. Employing the heat sink model, we show that the temporal growth rate for the linearised system is real, which allows us to perform a linear instability analysis. A nonlinear energy analysis is also presented, yielding a global stability threshold. We compare the heat sink model to the quadratic density model and find that the linearised systems are adjoint, implying that the instability boundaries which can be derived from the two models are the same. Received April 12, 2002 / Published online September 4, 2002 RID="a" ID="a" e-mail: Magdalen.Carr@durham.ac.uk RID="b" ID="b" e-mail: s.d.putter@tue.nl Communicated by Brian Straughan, Durham     

Nonsimilar Solutions for Mixed Convection in Non‐Newtonian Fluids Along a Vertical Plate in a Porous Medium

A nonsimilar boundary layer analysis is presented for the problem of mixed convection in powerlaw type nonNewtonian fluids along a vertical plate with powerlaw wall temperature distribution. The mixed convection regime is divided into two regions, namely,the forced convection dominated regime and the free convection dominated regime. The two solutions are matched. Numerical results are presented for the details of the velocity and temperature fields. A discussion is provided for the effect of viscosity index on the surface heat transfer rate.     

The Effect of Heavy Oil Viscosity Reduction by Solvent Dissolution on Natural Convection in the Boundary Layer of VAPEX

We have studied the effect of viscosity on natural convection in the boundary layer of the vapor extraction (VAPEX) process. VAPEX is a heavy oil recovery method that uses solvents to reduce oil viscosity, and is a potential process in reservoirs where thermal recovery methods cannot be applied. Natural convection may happen in VAPEX if the solvents that are used to decrease oil viscosity increase the density of the oil. This can especially occur with $\text{ CO }_{2}$ CO 2 -based solvents. Reduction of the oil viscosity due to solvent dissolution can have a large impact on the onset of convection by decreasing the critical Rayleigh number. When the viscosity reduction is significant, the critical Rayleigh can decrease up to two orders of magnitude. The transverse Peclet number is also a crucial parameter in determining the critical Rayleigh and onset of convection. Our analysis shows that the longitudinal Peclet does not have a significant impact on the natural convention in VAPEX. When oil viscosity reduction is included in the analysis of boundary layer instability in VAPEX, natural convection may occur in high-permeable reservoirs (where Rayleigh number is high) leading to a greater oil production rate compared with current models where the effect of boundary layer instability has been ignored.     

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.     

On the convective instability of a thermal boundary layer

When the surface temperature of a liquid is a harmonic function of time with a frequency, a temperature wave propagates into the liquid. The amplitude of this wave decreases exponentially with distance from the surface. The temperature oscillation is essentially concentrated in a layer of the order of (2/)^1/2, where x is the thermal conductivity of the liquid (thermal boundary layer). Depending on the phase, at certain positions below the surface the temperature gradient is directed downwards and if its magnitude is sufficiently large (the magnitude is a function of the amplitude and frequency of the surface oscillations) the liquid can become unstable with respect to the onset of convection. In that case the convective motion may spread beyond the initial unstable layer. For low frequencies the stability condition can be derived from the usual static Rayleigh criterion, on the basis of the Rayleigh number and the average temperature gradient of the unstable layer. This quasi-static approach, used by Sal'nikov [1], is appropriate to those cases in which the period of the temperature oscillations is much larger than the characteristic time of the perturbations. But when these times are of the same order, the problem must be analyzed in dynamic terms. The stability problem must then be formulated as a problem of parametricresonance excitation of velocity oscillations due to the action of a variable parameter-the temperature gradient.In an earlier work [2] we considered the problem of the stability of a horizontal layer of liquid with a periodically varying temperature gradient. It was assumed that the thickness of the layer was much smaller than the penetration depth of the thermal wave, so that the temperature gradient could be assumed to be independent of position. In the present work we consider the opposite case, in which the liquid layer is assumed to be much larger than the penetration depth, i. e., a thermal boundary layer can be defined. The temperature gradient at equilibrium, which is a parameter in the equations determining the onset of perturbations, is here a periodic function of time and a relatively complicated function of the depth coordinate z. The periodic oscillations are solved by the Fourier method; the equations for the amplitudes are solved by the approximate method of KarmanPohlhausen.The authors are grateful to L. G. Loitsyanskii for helpful criticism.     

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.     

Influence of Variable Permeability on Combined Convection Along a Nonisothermal Wedge in a Saturated Porous Medium

Mixed convection along a vertical nonisothermal wedge embedded in a fluid-saturated porous media incorporating the variation of permeability and thermal conductivity is studied. The surface temperature is assumed to vary as a power of the axial coordinate measured from the leading edge of the plate. A nonsimilar mixed convection parameter and a pseudo-similarity variable are introduced to cast the governing boundary layer equations into a system of dimensionless equations which are solved numerically using finite difference method. The entire mixed convection regime is covered by the single nonsimilarity parameter =[1+(Ra _x /Pe _x )^1/2]^–1 from pure forced convection (=1) to pure free convection (=0). The problem is solved using nonsimilarity solution for the case of variable wall temperature. Velocity and temperature profiles as well as local Nusselt number are presented. The wedge angle geometry parameter is ranged from 0 to 1.     

Mixed Convection-Radiation in Power-Law Fluids Along a NOn-Isothermal Wedge in a Porous Medium with Variable Permeability

A boundary layer analysis has been presented for the interaction of mixed convection with thermal radiation in laminar boundary flow from a vertical wedge in a porous medium saturated with a power-law type non-Newtonian incorporating the variation of permeability and thermal conductivity. The transformed conservation laws are solved numerically for the case of variable surface temperature conditions. The combined convection non-similar parameter we note that =0 and 1 correspond to pure free and forced convection cases. The Rosseland approximation is used to describe the radiative heat flux in energy equation. Velocity and temperature profiles as well as the local Nusselt number are presented.     

Three dimensional simulations of penetrative convection in a porous medium with internal heat sources

The problem of penetrative convection in a fluid saturated porous medium heated internally is analysed. The linear instability theory and nonlinear energy theory are derived and then tested using three dimensions simulation.Critical Rayleigh numbers are obtained numerically for the case of a uniform heat source in a layer with two fixed surfaces. The validity of both the linear instability and global nonlinear energy stability thresholds are tested using a three dimensional simulation. Our results show that the linear threshold accurately predicts the onset of instability in the basic steady state. However, the required time to arrive at the basic steady state increases significantly as the Rayleigh number tends to the linear threshold.     

Consequences of the Translation Invariance on the Darcy Free Convection Flow Past a Vertical Surface

It is shown that the governing equation for the stream function of the Darcy free convection boundary layer flows past a vertical surface is invariant under arbitrary translations of the transverse coordinate y. The consequences of this basic symmetry property on the solutions corresponding to a prescribed surface temperature distribution T _w (x) are investigated. It is found that starting with a “primary solution” which describes the temperature boundary layer on an impermeable surface, infinitely many “translated solutions” can be generated which form a continuous group, the “translation group” of the given primary solution. The elements of this group describe free convection boundary layer flows from permeable counterparts of the original surface with a transformed temperature distribution \({\tilde {T}_w \left( x \right)}\), when simultaneously a suitable lateral suction/injection of the fluid is applied. It turns out in this way that several exact solutions discovered during the latter few decades are in fact not basically new solutions, but translated counterparts of some formerly reported primary solutions. A few specific examples are discussed in detail.     

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.     

Hydrodynamics in weak force fields onset of steady thermocapillary convection in a spherical fluid layer under zero-g conditions

In the study of cellular convection in an infinite plane fluid layer with a free surface, both the Archimedes and thermocapillary forces [1–3] have been cited as reasons for the onset of convection. This has also been confirmed experimentally [4], When mass forces are absent or negligibly small it is natural to pose the question of the onset of pure thermocapillary convection or convection caused only by the surface tension gradients (see [2–3]). In the present paper, this problem is examined for a spherical fluid layer under zero-g conditions.     

The Effect of a Low-Frequency Structure on Passive Scalar Transport in the Flow Over a Surface-Mounted Rib

The unsteadiness of the flow over a surface-mounted rib involving passive scalar transport is studied using large-eddy simulation (LES) at a Reynolds number of 3000 (based on the rib height, \(h\), and the free-stream velocity, \(U_{0})\). The purpose of the present study is to gain further insight into the physical origin of the flow instability and its effect on passive scalar transport. Fourier spectral analysis of the velocity at different positions suggests that, in addition to the K-H instability in the shear layer (St ≈?0.42), two lower frequencies (St ≈?0.06 and 0.09) also exist. It is observed that the low-frequency instabilities accompany the shedding process of vortical structures. One low frequency, at \(\text {St}\approx 0.06\), is related to the pumping motion of the recirculation bubble, while the other, at \(\text {St}\approx 0.09\), is associated with the flapping mode of the shear layer. Through comparisons of velocity and temperature fields and the spectra of scalar fluctuations, it is found that the passive scalar is transported by the convection of vortical structures.     

Linear Stability of a Developing Thermal Front Induced by a Constant Heat Flux

A developing thermal front is set up by suddenly imposing a constant heat flux on the lower horizontal boundary of a semi-infinite fluid-saturated porous domain. The critical time for the onset of convection is determined using two main forms of analysis. The first of these is an approximate method which is effectively a frozen-time model while the second implements a set of parabolic simulations of monochromatic disturbances placed in the boundary layer at an early time. Results from the two approaches are compared and it is found that instability only occurs when the nondimensional disturbance wavenumber, $k$ k , is less than unity. The neutral curve for the primary mode possesses a vertical asymptote at $k=1$ k = 1 in wavenumber/time parameter space which is in contrast to the more usual teardrop shape which occurs when the surface is subject to a constant temperature. Asymptotic analyses are performed for the frozen-time model which yield excellent predictions for both branches of the neutral curve and the locus of the maximum growth rate curve at late times.     

Onset of Buoyancy-driven Instability in Porous Medium Solidified from Above

When a porous melt layer saturated by liquid is solidified from above, convection often sets in due to buoyancy forces. In this study, the onset of buoyancy-driven convection during time-dependent solidification is investigated by using the similarly transformed disturbance equations. The thermal disturbance distribution of the solid phase is approximated by the WKB method and effects of various parameters on the stability condition of the melt phase are analyzed theoretically. For the limiting case of λ → 0 and finite k _r, the critical conditions approach asymptotically and . This study presenting a constant-temperature cooling model predicts greater instability and gives more unstable results than those obtained from the constant solidification rate model.     

Weakly Nonlinear Stability Analysis of Temperature/Gravity-Modulated Stationary Rayleigh–Bénard Convection in a Rotating Porous Medium

The effect of time-periodic temperature/gravity modulation on thermal instability in a fluid-saturated rotating porous layer has been investigated by performing a weakly nonlinear stability analysis. The disturbances are expanded in terms of power series of amplitude of convection. The Ginzburg–Landau equation for the stationary mode of convection is obtained and consequently the individual effect of temperature/gravity modulation on heat transport has been investigated. Further, the effect of various parameters on heat transport has been analyzed and depicted graphically.     

Unsteady Thermo-Gravitational Convection Effects in a Side-Heated or Cooled Near-Critical Fluid

Unsteady thermo-gravitational convection of a near-critical fluid in an enclosed cavity, whose side wall temperature increases or decreases while the other boundaries are thermally insulated, is considered. An effective numerical method based on a two-scale pressure representation is used for solving the complete Navier-Stokes equations with a Van-der-Waals equation of state. For the neighborhood of the critical point, a transformation of the similarity parameters, which allows the introduction of effective values of these parameters, is found. The characteristic times of rapid temperature equalization due to adiabatic compression (piston effect), heat conduction, and thermo-gravitational convection are compared. The reasons why, in an unsteady convective jet, the temperature of the near-critical fluid is higher than the fixed side-wall temperature are analyzed.     

Convection in a two-layer system due to the combined action of the Rayleigh and thermocapillary instability mechanisms

In a two-layer system loss of stability may be monotonic or oscillatory in character. Increasing oscillatory perturbations have been detected in the case of both Rayleigh [1, 2] and thermocapillary convection [3–5]; however, for many systems the minimum of the neutral curve corresponds to monotonic perturbations. In [5] an example was given of a system for which oscillatory instability is most dangerous when the thermogravitational and thermocapillary instability mechanisms are simultaneously operative. In this paper the occurrence of convection in a two-layer system due to the combined action of the Rayleigh (volume) and thermocapillary (surface) instability mechanisms is systematically investigated. It is shown that when the Rayleigh mechanism operates primarily in the upper layer of fluid, in the presence of a thermocapillary effect oscillatory instability may be the more dangerous. If thermogravitational convection is excited in the lower layer of fluid, the instability will be monotonic.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 166–170, January–February, 1987.