Box Aspect Ratio Research Articles

Long-term nonlinear behaviour of the magnetorotational instability in a localized model of an accretion disc

For more than a decade, the so-called shearing box model has been used to study the fundamental local dynamics of accretion discs. This approach has proved to be very useful because it allows high resolution and long term studies to be carried out, studies that would not be possible for a global disc. Localised disc studies have largely focused on examining the rate of enhanced transport of angular momentum, essentially a sum of the Reynolds and Maxwell stresses. The dominant radial-azimuthal component of this stress tensor is, in the classic Shakura-Sunayaev model, expressed as a constant alpha times the pressure. Previous studies have estimated alpha based on a modest number of orbital times. Here we use much longer baselines, and perform a cumulative average for alpha. Great care must be exercised when trying to extract numerical alpha values from simulations: dissipation scales, computational box aspect ratio, and even numerical algorithms all affect the result. This study suggests that estimating alpha becomes more, not less, difficult as computational power increases.

Box-size dependence and breaking of translational invariance in the velocity statistics computed from three-dimensional turbulent Kolmogorov flows

The Kolmogorov flow generated by a stationary one-dimensional forcing varying sinusoidally in space is studied using direct numerical simulations with periodic boundary conditions. The velocity statistics are considered for various computational domains ranging from the minimal cubic box with size equal to the forcing wavelength to significantly larger domains. For the minimal computational domain, the velocity statistics exhibit symmetries that are directly imposed by the forcing properties. However, for larger domains, the translational invariance in the streamwise direction appears to be broken and the turbulence statistics depend on the computational box aspect ratio.

A numerical examination of the direct shear test

The paper presents results of two-dimensional simulations of the direct shear test using the discrete element method (DEM). The heterogeneous distributions of stress and strain are illustrated and, taking advantage of the numerical technique, the evolution of the stress tensor and porosity changes in the shear zone are monitored. The results demonstrate that the dilation inside the shear zone is much greater than that deduced from boundary measurements. The stress ratio τ/σ calculated from the boundary forces is about 10% greater than that calculated in the shear zone. It is also demonstrated that coaxiality of stress and strain rate exists at the critical state. Further simulations show that the influence of wall friction along the vertical faces of the specimen can be minimised by adopting the ‘symmetrical arrangement’ suggested by Jewell, and that the actual stress–strain–dilation behaviour within the shear zone is unaffected by the detailed boundary conditions. Comparison with previously published simulations results suggests that, in laboratory experiments, more reliable data, in the context of both stress ratio and dilation, would be obtained if the shear box aspect ratio was reduced below the value of 1:2 adopted in the traditional Casagrande shear box.

Construction of a Dynamical System Governing the Rayleigh–Bénard Convection in a Rectangular Box

A dynamical system governing the Rayleigh–Benard convection in a rectangular box is constructed with the aid of the Galerkin method starting from a Boussinesq system consisting of the Navier–Stokes and heat-transport equations. The boundary conditions are such that the velocity satisfies the rigid conditions on all the wall surfaces; and that the temperature satisfies the isothermal and insulating conditions on the horizontal and lateral wall surfaces respectively. The resulting dynamical system is used to evaluate critical Rayleigh numbers for onset of convection and those for oscillatory instability each together with the associated spatial patterns of the emerging convection fields under the conditions of box aspect ratios of Γ x =3.5, Γ y =2.0 and a Prandtl number of Pr =2.5. The results are shown to be in fairly good agreements with experiments available at present and other computations.

Nonlinear convective stability problems of viscoelastic fluids in finite domains

A Chebyshev pseudospectral method is generalized to solve the nonlinear hydrodynamic stability problems of Rayleigh-Benard convection of viscoelastic fluids in finite domains, which are compatible with the experimental situations, for the range of viscoelastic parameters where the exchange of stabilities is valid. The effects of box aspect ratio, the Deborah number λ and the dimensionless retardation time e on the critical Rayleigh number and convection intensity are investigated. The comparison of these results with the experimental data might be used to guide the selection of constitutive equations and to estimate viscoelastic parameter values. The present technique of hydrodynamic stability analysis is quite versatile and can be employed to solve other hydrodynamic stability problems in finite domains.

Parametrization of heat transport by non-Newtonian convection

SUMMARY We study the heat transport e⁄ciency of the 2-D steady convection in a box (aspect ratio~1) with a power-law creep which may be important for mantle convection. Eiects of pressure- (depth) and temperature-dependent creep are also studied. To analyse the heat transport, we use the local Rayleigh (Ral) and Nusselt (Nul) numbers, which are de¢ned by the values characterizing each thermal boundary layer except the length scale. The commonly used de¢nition of the Rayleigh and Nusselt numbers is only useful when the top and bottom thermal boundary layers show similar behaviour. Our de¢nition has the advantage of treating each thermal boundary layer separately. By choosing an appropriate temperature drop and viscosity, we ¢nd that the Ral^Nul relation of non-Newtonian £uid is in good agreement with that of Newtonian convection with a constant viscosity for most cases. Generally, the viscosity weighted by the strain rate is an appropriate viscosity for the range of Rayleigh number (10 4 *10 7 ) studied. However, it appears that this weighting becomes unsuitable at higher Rayleigh numbers. Strongly temperature-dependentcreep

A Solution Method of Nonlinear Convective Stability Problems in Finite Domains

A Chebyshev pseudospectral method is generalized to solve the linear and nonlinear hydrodynamic stability problems of thermal convection in a two-dimensional rectangular box with rigid sidewalls, where there may exist a heat source or a magnetic field to enhance or suppress the convection. The incompressibility condition is imposed rigorously on all boundaries. The effects of box aspect ratio, heat source, and magnetic field on the critical Rayleigh number and convection cell size are examined and compared with the results of other investigators. We have extended the present technique to nonlinear stability analysis and derived the Landau equation that describes the temporal evolution of the strength of convection in the rectangular box with rigid sidewalls. The results of nonlinear stability analysis are compared with the exact results obtained by the numerical solution of the Boussinesq equation. The present technique solves linear and nonlinear convective stability problems accurately and can be employed to solve other hydrodynamic stability problems in finite domains.

Rayleigh–Bénard convection of viscoelastic fluids in finite domains

We investigate the linear stability problem of the Rayleigh–Bénard convection of viscoelastic fluids in a two-dimensional rectangular box with nonslip sidewalls, where there may exist a heat source to enhance or suppress the convection. A Chebyshev pseudospectral method is generalized to solve the hydrodynamic stability problem. We adopt a very general constitutive equation that encompasses the Maxwell model, the Oldroyd model and the Phan–Thien–Tanner model. The effects of box aspect ratio, heat source, the Deborah number λ and the dimensionless retardation time ϵ on the critical Rayleigh number and convection cell size are examined. The range of λ and the ϵ for the onset of overstability is also obtained for a given box aspect ratio. The results of the present paper may be used to investigate the appropriateness of a constitutive equation and its parameter values adopted for a given viscoelastic fluid.

The Number of Hotspots in Mantle Convection: Effect of Depth-Dependent Viscosity and Internal Heating in Two-Dimensional Models

Two-dimensional numerical models of mantle convection have been calculated for a range of high Rayleigh numbers, depth-dependent viscosity and basal plus internal heating. Large aspect ratio boxes have been used in the calculations in order to estimate the expected areal density of upwellings in infinite fluid layers. The results are analyzed with regard to the number of the Earth’s hotspots which are assumed to be surface imprints of cylindrical upwellings in the mantle. For a pure whole-mantle situation, 6–7 upwellings can be expected. If the upper mantle convects separately above the 660 km discontinuity (allowing a second convective layer below 660 km depth), the theoretically estimated number of upper-mantle plumes can be as high as 250. Given the number of real hotspots (42 to 117 according to different compilations), it is suggested that the flow regime of the mantle is intermediate between the pure whole-mantle or pure two-layer circulation.

Transitions in thermal convection with strongly temperature-dependent viscosity in a wide box

Numerical models are systematically presented for time-dependent thermal convection of Newtonian fluid with strongly temperature-dependent viscosity in a two-dimensional rectangular box of aspect ratio 3 at various values of the Rayleigh number Ra b defined with viscosity at the bottom boundary up to 1.6×10 8 and the viscosity contrast across the box r η up to 10 8. We found that there are two different series of bifurcations that take place as r η increases. One series of bifurcations causes changes in the behavior of the thermal boundary layer along the surface boundary from small-viscosity-contrast (SVC) mode, through transitional (TR) mode, to stagnant-lid (ST) mode, or from SVC mode directly to ST mode, depending on Ra b. Another series of bifurcations causes changes in the aspect ratio of convection cells; convection with an elongated cell can take place at moderate r η (10 3–10 5.5 at Ra b=6×10 6), while only convection of aspect ratio close to 1 takes place at small r η and large r η. The parameter range of r η and Ra b for elongated-cell convection overlaps the parameter range for SVC and ST modes and include the entire parameter range for TR mode. In the elongated-ST regime, the lid of highly viscous fluid along the top boundary is not literally ‘stagnant’ but can horizontally move at a velocity high enough to induce a convection cell with aspect ratio much larger than 1.

Transition to Chaos in the Rayleigh-Bénard Convection

Direct numerical simulations of the Rayleigh-Benard convection in a rectangular box are run using the MAC method at the assigned values of the external parameters such that the Prandtl number P r = 5 and the box aspect ratios Γ x =3.5 and Γ y =2. The grid consists of 84 ×48 ×24 uniformly divided cubic cells. The results show that with increase of the Rayleigh number the oscillatory motion undergoes successive transitions in such a way as mono-periodic → bi-periodic → frequency-locked state → chaotic, which are in favorable agreement with the available experimental results due to Gollub and Benson obtained under the same settup of the external parameters P r , Γ x , and Γ y .

The timescales of plume generation caused by continental aggregation

To understand the thermal evolution of the mantle following the aggregation of non-subductable thick continental lithosphere, we study a numerical model in which a supercontinent, simulated by high viscosity raft, HVR, covers a part of the top surface of a convection layer. We model infinite Prandtl number convection either in a three-dimensional (3D) spherical shell, 3D rectangular box (aspect ratios: 8 and 4) or two-dimensional (2D) rectangular box (aspect ratio: 8) and except for the HVR, we specify a constant viscosity. The HVR, which has a viscosity higher than that of its surrounding, is instantaneously placed on the top surface of a well-developed convection layer and its position is fixed. Our results from 3D spherical shell cases with and without phase transitions show the emergence of a large plume characterized by a long wavelength thermal anomaly (a degree one pattern) for a Pangea-like geometry. We analyze the volume averaged temperature under the HVR (=〈 T C〉) the remaining (oceanic) area (=〈 T O〉) and total area (=〈 T M〉) to determine the timescale of plume generation. The difference between 〈 T C〉 and 〈 T O〉(=Δ T CO) and 〈 T M〉 show the existence of two characteristic timescales.Δ T CO exhibits an initial rapid increase and may become constant or continue to gradually increase. Meanwhile, 〈 T M〉 shows a similar behavior but with a longer timescale. We find that these timescales associated with the increase of Δ T CO and 〈 T M〉 can be attributed to the formation of large scale flow (i.e. plume) and response of the whole system to the emplacement of the HVR, respectively. For 3D spherical cases, we find that the timescale of plume generation is 1–2 Gyr, if the Rayleigh number is 10 6. To determine the effects of the viscosity of the HVR, 2D versus 3D modeling and the effects of the internal heating, we have also studied 2D and 3D rectangular box cases. A factor of about two variation exists in the timescale of plume generation. It appears that the timescale becomes greater for a smaller amount of internal heating. This may be attributed to the time-dependent flow caused by the internal heating. For 2D cases, we find that the timescale of the high Rayleigh number (10 7) case is shortened by a factor of three to five when compared to the Ra=10 6 case, which is consistent with the simple boundary layer theory. This may imply that well-developed plumes may arise with the timescale of 0.2 to 0.4 Gyr (for Ra=10 7).

Experimental study of Rayleigh–Benard convection at intermediate Rayleigh numbers using interferometric tomography

An experimental study of Rayleigh–Benard convection in an intermediate aspect ratio box that is square in plan is reported. An intermediate range of Rayleigh numbers has been considered in the study. The fluid employed is air. A Mach–Zehnder interferometer is used to collect the line-of-sight projections of the temperature field in the form of interferometric fringes. Images have been recorded after a sufficient time has elapsed for the initial transients to have been eliminated. Interferograms have been collected from four to six view angles. These are used to obtain the three dimensional temperature field inside the cavity by using tomography. An algebraic reconstruction technique has been used for the inversion of the projection data. The convergence of the iterative inversion procedure was unambiguous and asymptotic. The reconstructed temperature field with a subset of the total data was found to be consistent with the remaining unused projections. Results for two Rayleigh numbers, namely 13 900 and 40 200 have been reported. These were found to correspond to two distinct flow regimes. At these Rayleigh numbers, a well-defined steady state was not observed. At the lower Rayleigh number, the fringes away from the wall showed mild unsteadiness. At the higher Rayleigh number, the fringes were found to switch between two patterns. Results for the dominant mode alone have been presented for this problem. At a Rayleigh number of 13 900, three dimensional flow structures, whose influence is equivalent to longitudinal rolls have been observed. At a Rayleigh number of 40 200, cubic cells have been noted in the cavity. The associated flow pattern is inferred to be a plume rising from the heated plate. The local Nusselt number variation is seen to be consistent with the observed flow patterns.

Effects of a realistic mantle thermal conductivity on the patterns of 3-D convection

A physical model describing the temperature and pressure dependences of mantle thermal conductivity has recently been provided by A.M. Hofmeister [Science 283 (1999) 1699–1706]. We have studied numerically the influences of such a temperature- and pressure-dependent thermal conductivity on 3-D constant viscosity convection. A large aspect-ratio box of 4 × 4 × 1 has been taken for Rayleigh numbers greater than 104 to 106. The power-law index governing the phonon contribution to the thermal conductivity controls the style of convection. For silicates, the pattern of convection is quite different from that of constant conductivity, whereas the pattern of convection associated with conductivity of oxides is close to the case for constant conductivity. Both Boussinesq and extended-Boussinesq models with viscous and adiabatic heatings have been examined. Major effects of variable thermal conductivity on mantle convection are: (1) heating up of the lower mantle; (2) longer-wavelength Boussinesq convection; (3) shorter wavelength for extended-Boussinesq convection; (4) thick stable plumes with large plumeheads; (5) thicker thermal boundary layer of around 500 km at the base of the mantle. The contributions from the new terms due to the variable conductivity in the energy equation have magnitudes several times that of chondritic heating at the boundary layers. A large region of superadiabatic lower mantle with about a 400 to 600 K excess is caused by variable thermal conductivity. This finding has an interesting implication on the recent seismic finding of a lighter density in the deep portion of the lower mantle.

Effect of Surface Corrugation on Convection in a Three-Dimensional Finite Box of Fluid Saturated Porous Material

The problem of finite amplitude thermal convection in a three-dimensional finite box of fluid saturated porous material is investigated, when the lower boundary of the fluid is corrugated. The nonlinear problem of three-dimensional convection in the box for the values of the Rayleigh number close to the classical critical value and for small values of the amplitude of the corrugations is solved by a perturbation technique. The preferred mode of convection is determined by stability analysis. In the absence of corrugation three-dimensional modes of convection can be either stable or unstable depending on the values of the aspect ratios of the box, while two-dimensional rolls are always stable, provided that the box aspect ratios allow the existence of such modes of convection. In the presence of boundary corrugation with the appropriate form, different three-dimensional or two-dimensional modes of corrugation can be stable or unstable. For a rough boundary with local roughness sites, the location, size, and number of the roughness elements plus the wave numbers of the convection modes and the box aspect ratios can all play a role leading to either stable or unstable particular three- or two-dimensional flow patterns. For a wavy boundary, resonant wave-vector excitation can lead to the preference of stable two- or three-dimensional flow patterns whose wave vectors are in a subset of those due to the wavy boundary, while nonresonant wave-vector excitation can lead to the preference of stable flow patterns whose wave vectors are not generally in a subset of those due to the wavy boundary. Heat transported by convection can either be enhanced or be reduced by certain proper forms of the corrugations and by appropriate values of the box aspect ratios. Due to the surface corrugation highly subcritical modes of convection are stable, while highly supercritical modes of convection are unstable.

Further Convergence Studies of the Enhanced Doublet-Lattice Method

The doublet-lattice method (DLM) has become a standard source for the aerodynamic loads in  utter and dynamic response analyses of aircraft at subsonic speeds. A recent reŽ nement to the method is based on an improvement to the integration of the oscillatory kernel function by replacing the parabolic approximation in the kernel numeratorwith amore accurate quartic approximation.This reŽ nement extends the frequency limits of the DLM for applications to higher-frequency  utter and dynamic response. The effects of Mach number and lattice geometry on convergence by numerical experimentationwith some previously studied conŽ gurations are reviewed as well as a realistic aircraft model, present guidelines are reviewed, and updated guidelines for box aspect ratio and number of boxes per wavelength are presented.

RAYLEIGH-BENARD CONVECTION IN LIMITED DOMAINS: PART 1 - OSCILLATORY FLOW

Transition from the steady state to an oscillatory regime in three-dimensional limited aspect ratio boxes, filled with an incompressible fluid and heated from below, has been examined by direct numerical simulation. Two different physical problems have been considered: the first is related to a domain 3.5 x 1 x 2.1 filled with water at 70 degrees C (Prandtl number 2.5); the second considers a domain 2.4 x 1 x 1.2 filled with water at 33 degrees C (Prandtl number 5). The Rayleigh number has been varied from 20,000 to 80,000. A new procedure based on a statistical approach for evaluation of the critical Rayleigh number for transition from steady state to oscillatory flow (RaII) has been introduced in order to reduce numerical errors and estimate the error bars. A systematic study for the determination of RaII has been conducted as a function of the geometries considered and the different flow structures observed.

RAYLEIGH-BENARD CONVECTION IN LIMITED DOMAINS: PART 2 - TRANSITION TO CHAOS

Transitions to chaos in three-dimensional limited aspect ratio boxes, filled with an incompressible fluid and heated from below, have been examined by direct numerical simulation as the Rayleigh number is varied. Two different problems have been considered: the first is related to a domain 3.5 x 1 x 2.1 filled with water at 70 degrees C (Prandtl number 2.5); the second is related to a domain 2.4 x 1 x 1.2 filled with water at 33 degrees C (Prandtl number 5). The Rayleigh number has been varied from 45,000 up to 300,000. Three different bifurcation sequences have been detected, but only two individual mechanisms for the transition to the nonperiodic motion have been identified: the subharmonic cascade and the quasi-periodicity with three incommensurate frequencies. Effects of different regimes and flow structures on heat transfer have been discussed.

Further Refinement of the Subsonic Doublet-Lattice Method

The doublet-lattice method (DLM) is in use worldwide for e utter and dynamic response analyses of aircraft at subsonic speeds. The present paper develops a further ree nement to extend its frequency limits for applications to higher frequency e utter, e.g., for aeroservoelastic systems with high-frequency control surfaces, and dynamic response, e.g., for short wavelength gusts. The DLM is an aerodynamic e nite element method for modeling oscillating interfering lifting surfaces in subsonic e ows. It reduces to the vortex-lattice method at zero-reduced frequency. The number of e nite elements (boxes) required for accurate results depends on aspect ratio and reduced frequency, among other parameters. At high reduced frequency, the chordwise dimension of the boxes must be small. However, an approximation in the method, viz., that the variation of the numerator of the incremental oscillatory kernel function is parabolic across the span of the box bound vortex, restricts the box aspect ratio to about 3. Hence, high-frequency requirements bring an associated requirement for a large number of boxes in the aerodynamic ideali- zation. If a higher-order approximation is used for the spanwise variation of the numerator of the incre- mental oscillatory kernel, the limitation on box aspect ratio can be relaxed and the number of spanwise divisions required in high-frequency analyses will be reduced signie cantly, leading to a reduction in the total number of boxes. This paper replaces the original parabolic approximation by a quartic approxi- mation. The quartic curve-e tting coefe cients are determined for the planar and nonplanar kernels, and the new integrals for the planar and nonplanar normalwash factors are evaluated. The ree nement is incorporated into a DLM code previously known as N5KA, and convergence studies on typical cone gu- rations are presented that attempt to specify a higher limit for practical box aspect ratios.

Dynamics of strongly time-dependent convection with non-Newtonian temperature-dependent viscosity

Numerical simulations of time-dependent Boussinesq convection with a non-Newtonian temperature-dependent rheology have been conducted in wide aspect-ratio boxes for effective Rayleigh numbers, Ra v, ranging from 3.5 × 10 4 to 1.3 × 10 6. The transition to the chaotic regime for non-Newtonian temperature-dependent viscosity takes place at Ra v an order of magnitude lower than for purely temperature-dependent viscosity. For viscosity contrasts owing to temperature of O(10 2), lateral viscosity contrasts generated by the flow are typically of O(10 3) in the interior with contrasts as large as O(10 6) in the upper boundary layer. The horizontal Fourier spectra of the temperature field and of the inverse of the viscosity field show that the asymmetry in the flow introduced by a temperature-dependent viscosity is weaker for a non-Newtonian rheology than for a Newtonian rheology. Lithospheric dynamics, such as lubrication of subduction and lithospheric thinning, are greatly facilitated by the effects of stress-softening from non-Newtonian rheology. Weak plumes are slowed down drastically in the middle of their ascent because of the stress-dependent nature of the rheology. Downgoing slabs can be found to be weakened near the middle, and detachment involving pieces of the descending material may take place. The mobility of the stiff upper boundary layer is facilitated by an increase in the vigor of convection, whereas an increase in the viscosity contrast owing to temperature retards the surface mobility.