Basic Steady State Research Articles

Three-dimensional simulation for problem of penetrative convection near the maximum density

The problem of penetrative convection with the cubic and fifth-order equations of state proposed by Merker, Waas and Grigull is studied. Both linear instability and nonlinear stability analyses are performed to assess the suitability of linear theory to predict destabilisation of the convection. The validity of both the linear instability and global nonlinear energy stability thresholds are tested using three-dimensional simulation. The results show that the linear threshold accurately predicts the onset of instability in the basic steady state solution.

Stability analysis of penetrative convection in anisotropic porous media with variable permeability

Abstract In this paper we study the penetrative convection in an anisotropic porous medium, for fluids with quadratic density law with the anisotropic permeability and thermal diffusivity effects. Both linear instability and global non-linear energy stability analyses are tested using a three-dimensional simulation. Our results show that the linear threshold accurately predicts on the onset of instability in the basic steady state. However, the required time to arrive at the steady state increases significantly as the Rayleigh number tends to the linear threshold.

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.

Global instabilities of lamellar eutectic growth in directional solidification

The present paper is concerned with the global instability mechanisms for the basic steady state of lamellar eutectic growth with curved and tilted interface affected by the triple point. We solve the related linear eigenvalue problem (EVP) by using the analytical approach for the case that the Peclet number is small and the segregation coefficient parameter κ is close to the unit. It is found that the system involves two types of global instability mechanisms: the ‘exchange of stability’ invoked by the non-oscillatory, unstable modes and the global wave instabilities invoked by four types of oscillatory unstable modes, namely (AA)-, (SS)-, (AS)- and (SA)-mode. The quantization conditions for these modes are derived, the neutral curves for these global instabilities on the parameters plane are calculated. The stability criterion yields the range of the interlamellar spacing and explains the interfacial pattern transitions observed in the experiments.

Secondary flows in a laterally heated horizontal cylinder

In this paper we study the problem of thermal convection in a laterally heated, finite, horizontal cylinder. We consider cylinders of moderate aspect ratio (height/diameter ≈2) containing a small Prandtl number fluid (σ < 0.026) typical of molten metals and molten semiconductors. We use the Navier-Stokes and energy equations in the Boussinesq approximation to calculate numerically the basic steady states, analyze their linear stability, and compute some nonlinear secondary flows originated from the instabilities. All the calculated flows and the stability analysis are characterized by their symmetry properties. Due to the confined cylindrical geometry, -presence of lateral walls and lids-, all the flows are completely three dimensional even for the basic steady states. In the range of Prandtl numbers studied, we have identified four different types of instabilities, either oscillatory or stationary. The physical mechanisms, shear or buoyancy, of the corresponding flow transitions have been analyzed. As the value of the Prandtl number approaches σ = 0.026 the scenario of bifurcations becomes more complicated due to the existence of two different stable basic states originated in a saddle-node bifurcation; a fact that had been overlooked in previous works.

Three-Dimensional Simulations for Convection Problem in Anisotropic Porous Media with Nonhomogeneous Porosity, Thermal Diffusivity, and Variable Gravity Effects

A convection problem in anisotropic and inhomogeneous porous media has been analyzed. In particular, the effect of variable permeability, thermal diffusivity, and variable gravity with respect to the vertical direction, has been studied. A linear and nonlinear stability analysis of the conduction solution has been performed. 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 on 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.

Three dimensions simulation for the problem of a layer of non-Boussinesq fluid heated internally with prescribed heat flux on the lower boundary and constant temperature upper surface

The purpose of this paper is to study the effect of a heat source on the solution to the equations for an incompressible heat conducting viscous fluid. The validity of both the linear instability and global nonlinear energy stability thresholds are tested using three dimensional simulation. Our results show that the linear threshold accurately predicts on the onset of instability in the basic steady state. However, the required time to arrive at the steady state increases significantly as the Rayleigh number tends to the linear threshold.

Slow–fast dynamics of a time-delayed electro-optic oscillator

Square-wave oscillations exhibiting different plateau lengths have been observed experimentally by investigating an electro-optic oscillator. In a previous study, we analysed the model delay differential equations and determined an asymptotic approximation of the two plateaus. In this paper, we concentrate on the fast transition layers between plateaus and show how they contribute to the total period. We also investigate the bifurcation diagram of all possible stable solutions. We show that the square waves emerge from the first Hopf bifurcation of the basic steady state and that they may coexist with stable low-frequency periodic oscillations for the same value of the control parameter.

Formation mechanism and stability analysis of the Houba expansive soil landslide

To uncover the formation mechanism of the Houba landslide in the Three Gorges Reservoir Area in China, macroscopic deformation characteristics, matter and structural characteristics, and slope stability were investigated using geological field investigations, indoor and outdoor tests, mineral identification, etc. The results show that since a large deformation occurred in the Houba landslide in 2003, there has been no obvious overall deformation and the whole slope has been in a basic steady state since January 2004. Slope deformation mainly exists in the expansive soil layer from 0 to 5m (surface-layer sliding zone), and there are some evidences suggesting activity at 20m (shallow-layer sliding zone) and nearby 35m (deep-layer sliding zone). The expansive soil in the surface layer has a thickness of approximately 5m, is 25–35% chlorite and illite, and has a free swelling ratio of 25–80%, making it a weak-moderate expansive soil. The expansive soil also has strong hydrophilicity, and it is more sensitive to saturation changes. Its shear strength could be easily reduced by saturation–unsaturation cycles with the change of water table or rainfall, so the expansive soil itself creates a good condition for a gently inclined landslide. The surface-layer slope, shallow-layer slope and deep-layer slope are in steady states under natural and rainstorm conditions. However, the surface-layer landslide body on the back edge of the III–III′ profile is in a basic stable–unstable state under rainstorm conditions, and a surface-layer landslide may occur under continuous rainfall conditions. Many people live on the Houba landslide, and the coupling effect of expansive soil and rainfall could cause many more local landslides from the surface-layer expansive soil with a slope-free face. Some measures should be taken to control the landslide, such as surface draining, local landslide supporting and retaining, and reinforcing the front existing retaining wall.

Study of dynamics of glucose-glucose oxidase-ferricyanide reaction

This work is focused on dynamics of the glucose-glucose oxidase-ferricyanide enzymatic reaction with or without sodium hydroxide in a continuous-flow stirred tank reactor (CSTR) and in a batch reactor. This reaction exhibits pH-variations having autocatalytic character and is reported to provide nonlinear dynamic behavior (bistability, excitability). The dynamical behavior of the reaction was examined within a wide range of inlet parameters. The main inlet parameters were the ratio of concentrations of sodium hydroxide and ferricyanide and the flow rate. In a batch reactor we observed an autocatalytic drop of pH from slightly basic to medium acidic values. In a CSTR our aim was to find bistability in the presence of sodium hydroxide. However, only a basic steady state was found. In order to reach an acidic steady state, we investigated the system in the absence of sodium hydroxide. Under these conditions the transition from the basic to the acidic steady state was observed when inlet glucose concentration was increased.

Anisotropic turbulent thermal diffusion and thermal convection in a rapidly rotating fluid sphere

Estimates of the molecular values of magnetic, viscous and thermal diffusion suggest that the state of the Earth’s core is turbulent and that complete numerical simulation of the geodynamo is not realizable at present. Large eddy simulation of the geodynamo with modelling of the sub-grid scale turbulence must be used. Current geodynamo models effectively model the sub-grid scale turbulence with isotropic diffusivities larger than the molecular values appropriate for the core. In the Braginsky and Meytlis (1990) picture of core turbulence the thermal and viscous diffusivities are enhanced up to the molecular magnetic diffusivity in the directions of the rotation axis and mean magnetic field. We neglect the mean magnetic field herein to isolate the effects of anisotropic thermal diffusion, enhanced or diminished along the rotation axis, and explore the instability of a steady conductive basic state with zero mean flow in the Boussinesq approximation. This state is found to be more stable (less stable) as the thermal diffusion parallel to the rotation axis is increased (decreased), if the transverse thermal diffusion is fixed. To examine the effect of simultaneously varying the diffusion along and transverse to the rotation axis, the Frobenius norm is used to control for the total thermal diffusion. When the Frobenius norm of the thermal diffusion tensor is fixed, it is found that increasing the thermal diffusion parallel to the rotation axis is destabilising. This result suggests that, for a fixed total thermal diffusion, geodynamo codes with anisotropic thermal diffusion may operate at lower modified Rayleigh numbers.

Aspects of a hybrid hp-finite element/spectral method for the linearised magnetohydrodynamic equations in spherical geometries

We consider the incompressible magnetohydrodynamic equations, linearised about a steady basic state in spherical geometries. The angular dependence is discretised using a Galerkin method based on spherical harmonics. This produces a coupled system of ordinary differential equations in radius, which may contain up to fourth order derivatives, after separation of the time dependence. In applications small magnetic, viscous or thermal diffusion may lead to boundary layers of second or higher order. We investigate key aspects of the discretisation of the radial equations using one dimensional $hp$-finite element methods through two model problems: reduction of order versus higher order elements for an ordinary differential equation of fourth order; and fourth order boundary layers. Results indicate that higher order elements are to be favoured over a reduction of order, and that robust exponential convergence may be achieved for the symmetric two layer problem investigated.

Joint modelling of obstacle induced and mesoscale changes—Current limits and challenges

Obstacles considerably influence boundary layer processes. Their influences have been included in mesoscale models (MeM) for a long time. Methods used to parameterise obstacle effects in a MeM are summarised in this paper using results of the mesoscale model METRAS as examples. Besides the parameterisation of obstacle influences it is also possible to use a joint modelling approach to describe obstacle induced and mesoscale changes. Three different methods may be used for joint modelling approaches: The first method is a time-slice approach, where steady basic state profiles are used in an obstacle resolving microscale model (MiM, example model MITRAS) and diurnal cycles are derived by joining steady-state MITRAS results. The second joint modelling approach is one-way nesting, where the MeM results are used to initialise the MiM and to drive the boundary values of the MiM dependent on time. The third joint modelling approach is to apply multi-scale models or two-way nesting approaches, which include feedbacks from the MiM to the MeM. The advantages and disadvantages of the different approaches and remaining problems with joint Reynolds-averaged Navier–Stokes modelling approaches are summarised in the paper.

Optimal harmonic response in a confined Bödewadt boundary layer flow

The Bödewadt boundary layer flow on the stationary bottom end wall of a finite rotating cylinder is very sensitive to perturbations and noise. Axisymmetric radial waves propagating inward have been observed experimentally and numerically before the appearance of spiral three-dimensional instabilities. In this study, the sensitivity and response of the finite Bödewadt flow to a harmonic modulation of the rotation rate are analyzed. A comprehensive exploration of response to variations in the amplitude and frequency of the forcing has been carried out. There are sharply delineated linear- and nonlinear-response regimes, with a sharp transition between them at moderate amplitudes. The periodic forcing leads to a steady-streaming flow, even in the linear-response regime, and to a period-doubling bifurcation in the nonlinear regime. Frequency response curves at different forcing amplitudes over a wide range of frequencies have been computed and used to identify the frequency band that excites the axisymmetric radial waves and the forcing frequency that elicits the strongest response. Finally, we have shown that the axisymmetric waves always decay to the steady basic state when the harmonic modulation is suppressed, and conclude that the experimentally observed persistent circular waves are not self-sustained.

The Baroclinic Adjustment of Time-Dependent Shear Flows

Abstract Motivated by the fact that time-dependent currents are ubiquitous in the ocean, this work studies the two-layer Phillips model on the beta plane with baroclinic shear flows that are steady, periodic, or aperiodic in time to understand their nonlinear evolution better. When a linearly unstable basic state is slightly perturbed, the primary wave grows exponentially until nonlinear advection adjusts the growth. Even though for long time scales these nearly two-dimensional motions predominantly cascade energy to large scales, for relatively short times the wave–mean flow and wave–wave interactions cascade energy to smaller horizontal length scales. The authors demonstrate that the manner through which these mechanisms excite the harmonics depends significantly on the characteristics of the basic state. Time-dependent basic states can excite harmonics very rapidly in comparison to steady basic states. Moreover, in all the simulations of aperiodic baroclinic shear flows, the barotropic component of the primary wave continues to grow after the adjustment by the nonlinearities. Furthermore, the authors find that the correction to the zonal mean flow can be much larger when the basic state is aperiodic compared to the periodic or steady limits. Finally, even though time-dependent baroclinic shear on an f plane is linearly stable, the authors show that perturbations can grow algebraically in the linear regime because of the erratic variations in the aperiodic flow. Subsequently, baroclinicity adjusts the growing wave and creates a final state that is more energetic than the nonlinear adjustment of any of the unstable steady baroclinic shears that are considered.

Operating Experiences on Commercial Fuel Cell Units for Back Up Power Generation

Abstract Extensive residential demonstration programs and the needs for innovative and reliable back up power systems are driving the development and diffusion of small (<10 kWe) stationary fuel cell power systems. Low temperature polymer electrolyte fuel cell (PEFC) power systems are particularly suitable for back up and UPS applications due to short start up times, whereas for small cogenerative residential applications both PEFC and solid oxide fuel cells (SOFCs) are emerging as promising technologies. The technical and economical viabilities of fuel cell based systems have been already demonstrated for a few niche applications such as back up system with high autonomy. Nevertheless fuel cell technologies are not yet mature. Durability and reliability are of great concern and have to be specifically addressed. Real world experiences and extensive laboratory testing are paramount for the development of reliable products, as well as to harmonize and refine codes and standards required for the market entry. This paper presents and discusses the results of a 3 year experience on commercial PEFC 1 kWe units. Basic characterization, cycling, and steady state endurance testing data are analyzed herein with a focus on power system performance, reliability, and degradation issues. End user and system integrator testing approaches were applied. Power system response to load demand and electrical efficiency were measured following as much as possible the prescriptions of codes and standards. The influences of operating and environmental conditions on system efficiency were investigated as well. Positive results were achieved and, in particular, system availability resulted extremely high. Steady state long term endurance tests evidenced, however, critical durability and safety issues to be improved.

Linear instability analysis of low-pressure turbine flows

Three-dimensional linear BiGlobal instability of two-dimensional states over a periodic array of T-106/300 low-pressure turbine (LPT) blades is investigated for Reynolds numbers below 5000. The analyses are based on a high-order spectral/hpelement discretization using a hybrid mesh. Steady basic states are investigated by solution of the partial-derivative eigenvalue problem, while Floquet theory is used to analyse time-periodic flow set-up past the first bifurcation. The leading mode is associated with the wake and long-wavelength perturbations, while a second short-wavelength mode can be associated with the separation bubble at the trailing edge. The leading eigenvalues and Floquet multipliers of the LPT flow have been obtained in a range of spanwise wavenumbers. For the most general configuration all secondary modes were observed to be stable in the Reynolds number regime considered. When a single LPT blade with top to bottom periodicity is considered as a base flow, the imposed periodicity forces the wakes of adjacent blades to be synchronized. This enforced synchronization can produce a linear instability due to long-wavelength disturbances. However, relaxing the periodic restrictions is shown to remove this instability. A pseudo-spectrum analysis shows that the eigenvalues can become unstable due to the non-orthogonal properties of the eigenmodes. Three-dimensional direct numerical simulations confirm all perturbations identified herein. An optimum growth analysis based on singular-value decomposition identifies perturbations with energy growthsO(105).

Harmonically forced enclosed swirling flow

The response of steady state vortex flows in an enclosed circular cylinder driven by the harmonic modulation of the rotating end wall is investigated experimentally and numerically. Three dynamic regimes have been identified, with a continuous variation in forcing frequency between them. For very low forcing frequency, the synchronous flow approaches quasistatic adjustment, and for very large forcing frequencies the oscillations in the synchronous flow are localized in the boundary layers on the various cylinder walls. These localized wall oscillations drive the synchronous flow in the cylinder interior to the underlying axisymmetric steady basic state. The third regime occurs for forcing frequencies in the range of the most dangerous axisymmetric Hopf eigenfrequencies, with the 1:1 resonances leading to greatly enhanced oscillation amplitudes localized in the axis region where the flow manifests vortex breakdown recirculation zones.

Linear instability of broad baroclinic zones

Baroclinic energy conversions play a central role in mid-latitude synoptic-scale dynamics, drawing energy from the meridional temperature gradient and feeding it into the eddy dynamics. The physical mechanisms underlying these conversions can be investigated through the theory of small-amplitude instabilities growing on a steady basic state. In a recent study the author analysed the instability associated with a boundary temperature transition concentrated in a narrow front. This paper looks at the complementary problem of a broad baroclinic zone. The solution is applicable both to the Eady model and to the Phillips two-level quasi-geostrophic model. The reduction in the growth rate relative to an infinitely broad, homogeneous baroclinic mne is inversely proportional to the width of the baroclinic zone, as found in previous studies. The constant of proportionality is expressed in a simplified and more general form, related to the derivative with respect to zonal wave number of the complex phase speed of the homogeneous problem. If the flow is purely baroclinic the maximum growth rate is reduced compared with that of a homogeneous, infinitely broad frontal zone by a factor 1 - (2kmaxr)-1. where kmax is the wave number of the most unstable mode of the homogeneous problem and r is the width of the baroclonic zone. The width, r, is defined in terms of the second meridional derivative of the boundary temperature gradient evaluated at the maximum of the latter. Results in the literature give consistent, though less general, expressions for the growth rate, but various predictions for the structure of the waves. The numerical solutions show that as the jet width is reduced a new type of mode appears in which critical lines at the lid (tropopause) play an important role.

Van der Waals–like phase-separation instability of a driven granular gas in three dimensions

We show that the van der Waals-like phase-separation instability of a driven granular gas at zero gravity, previously investigated in two-dimensional settings, persists in three dimensions. We consider a monodisperse granular gas driven by a thermal wall of a three-dimensional rectangular container at zero gravity. The basic steady state of this system, as described by granular hydrodynamic equations, involves a denser and colder layer of granulate located at the wall opposite to the driving wall. When the inelastic energy loss is sufficiently high, the driven granular gas exhibits, in some range of average densities, negative compressibility in the directions parallel to the driving wall. When the lateral dimensions of the container are sufficiently large, the negative compressibility causes spontaneous symmetry breaking of the basic steady state and a phase separation instability. Event-driven molecular dynamics simulations confirm and complement our theoretical predictions.