Abdelfattah Zebib

Numerical simulations of three-dimensional thermal convection in a fluid with strongly temperature-dependent viscosity

Numerical calculations are presented for the steady three-dimensional structure of thermal convection of a fluid with strongly temperature-dependent viscosity in a bottom-heated rectangular box. Viscosity is assumed to depend on temperature T as exp ( --ET), where E is a constant ; viscosity variations across the box r (= exp (E)) as large as lo5 are considered. A stagnant layer or lid of highly viscous fluid develops in the uppermost coldest part of the top cold thermal boundary layer when r > rcl, where r = rcl = 1.18 x 103R,0~308 andR, is the Rayleigh number based on the viscosity at the top boundary. Three-dimensional convection occurs in a rectangular pattern beneath this stagnant lid. The planform consists of hot upwelling plumes at or near the centre of a rectangle, sheets of cold sinking fluid on the four sides, and cold sinking plume concentrations immersed in the sheets. A stagnant lid does not develop, i.e. convection involves all of the fluid in the box when r rc2 = 3.84 x 106R;1.36. The planform of the convection is rectangular with the coldest parts of the sinking fluid and the hottest part of the upwelling fluid occurring as plumes at the four corners and at the centre of the rectangle, respectively. Both hot uprising plumes and cold sinking plumes have sheet-like extensions, which become more well-developed as r increases. The whole-layer mode of convection occurs as two-dimensional rolls when r < min (rcl, rc2). The Nusselt number Nu depends on the viscosity at the top surface more strongly in the regime of whole-layer convection than in the regime of stagnant-lid convection. In the whole-layer convective regime, Nu depends more strongly on the viscosity at the top surface than on the viscosity at the bottom boundary.

High Marangoni number convection in a square cavity

Steady thermocapillary flow is examined in a square two‐dimensional cavity with a single free surface and differentially heated side walls. The numerical solutions are obtained with a finite difference method applied to a streamfunction‐temperature formulation. This work investigates the Prandtl number dependence, structure, and stability of high Marangoni number flow. It is found that the character of thermocapillary flow is highly sensitive to the value of the Prandtl number over a range of Marangoni numbers exceeding 1×105 for 1≤Pr≤50, the magnitude of the flow showing nonmonotonic dependence on the Marangoni number for Pr≤∼10. A complete structural analogy is observed between flow in a cavity driven by a moving lid and thermocapillary flow in the boundary layer limit, and it is found that all the solutions, spanning a wide range of Marangoni and Prandtl numbers, are linearly stable to a restricted class of disturbances.

Three-dimensional thermal convection in a spherical shell

Independent pseudo-spectral and Galerkin numerical codes are used to investigate three-dimensional infinite Prandtl number thermal convection of a Boussinesq fluid in a spherical shell with constant gravity and an inner to outer radius ratio equal to 0.55. The shell is heated entirely from below and has isothermal, stress-free boundaries. Nonlinear solutions are validated by comparing results from the two codes for an axisymmetric solution at Rayleigh number Ra = 14250 and three fully three-dimensional solutions at Ra = 2000, 3500 and 7000 (the onset of convection occurs at Ra = 712). In addition, the solutions are compared with the predictions of a slightly nonlinear analytic theory. The axisymmetric solution is equatorially symmetric and has two convection cells with upwelling at the poles. Two dominant planforms of convection exist for the three-dimensional solutions: a cubic pattern with six upwelling cylindrical plumes, and a tetrahedral pattern with four upwelling plumes. The cubic and tetrahedral patterns persist for Ra at least up to 70000. Time dependence does not occur for these solutions for Ra [les ] 70000, although for Ra > 35000 the solutions have a slow asymptotic approach to steady state. The horizontal and vertical structure of the velocity and temperature fields, and the global and three-dimensional heat flow characteristics of the various solutions are investigated for the two patterns up to Ra = 70000. For both patterns at all Ra , the maximum velocity and temperature anomalies are greater in the upwelling regions than in the downwelling ones and heat flow through the upwelling regions is almost an order of magnitude greater than the mean heat flow. The preferred mode of upwelling is cylindrical plumes which change their basic shape with depth. Downwelling occurs in the form of connected two-dimensional sheets that break up into a network of broad plumes in the lower part of the spherical shell. Finally, the stability of the two patterns to reversal of flow direction is tested and it is found that reversed solutions exist only for the tetrahedral pattern at low Ra .

Transitions in thermal convection with strongly variable viscosity

One of the most important material properties influencing the style of convection in the mantles of terrestrial planets is the extreme temperature-dependence of viscosity. Three-dimensional numerical convection calculations in a wide (8 × 8 × 1) cartesian box and in a spherical shell (ratio of inner to outer radius of 0.55, characteristic of terrestrial planets) both display two fundamental transitions as the viscosity contrast is progressively increased from unity to a factor of 105. These transitions not only mark changes in the style of deformation in the upper boundary layer from mobile-lid to sluggish-lid to stagnant-lid but also have dramatic effects on the style, planform, and horizontal length scales of convection in the entire domain. Vertical variations of viscosity are the most important for determining the horizontal length scales of the convective patterns while lateral viscosity variations play a role in shaping the relative structures of the upwelling and downwelling flows. Convection in Venus appears to be represented most closely by the sluggish-lid regime of convection, whereas the Earth, with plate tectonics, more closely resembles the mobile-lid style of convection. Forcing plate-like characteristics onto the convective flows in the form of imposed weak zones and prescribed surface velocities results in flow patterns dominated entirely by the form used to enforce the plate-like behavior and tells us little about why the mantle exhibits long-wavelength heterogeneity.

A chebyshev method for the solution of boundary value problems

Abstract An expansion procedure using the Chebyshev polynomials as base functions is proposed. The method yields more accurate results than either of the Galerkin or tau methods as indicated from solving the Orr-Sommerfeld equation for both the plane Poiseuille flow and the Blasius velocity profile. The Chebyshev approximation is also applied to resolve the radial dependence of the flow field for a circular cylinder or a sphere in a uniform flow.

Steady tetrahedral and cubic patterns of spherical shell convection with temperature-dependent viscosity

Steady thermal convection of an infinite Prandtl number, Boussinesq fluid with temperature-dependent viscosity is systematically examined in a three-dimensional, basally heated spherical shell with isothermal and stress-free boundaries. Convective flows exhibiting cubic (𝓁 2, 𝓂 = {0, 4}) and tetrahedral (𝓁 = 3, 𝓂 = 2) symmetry are generated with a finite-volume numerical model for various combinations of Rayleigh number Ra (defined with viscosity based on the average of the boundary temperatures) and viscosity contrast 𝓇μ (ratio of maximum to minimum viscosities). The range of Ra for which these symmetric flows in spherical geometry can be maintained in steady state is sharply reduced by even mild viscosity variations (𝓇μ ≤ 30), in contrast with analogous calculations in Cartesian geometry in which relatively simple, three-dimensional convective planforms remain steady for 𝓇μ ≈ 104. The mild viscosity contrasts employed place some solutions marginally in the sluggish-lid transition regime in Ra-𝓇μ parameter space. Global heat transfer, given by the Nusselt number N𝓊, is found to obey a single power law relation with Ra when Ra is scaled by its critical value. A power law of the form N𝓊 ∼ (Ra/Racrit)1/4 (Racrit is the minimum critical value of Ra for the onset of convection) is obtained, in agreement with previous results for isoviscous spherical shell convection with cubic and tetrahedral symmetry. The calculations of this paper demonstrate that temperature-dependent viscosity exerts a strong control on the nature of three-dimensional convection in spherical geometry, an effect that is likely to be even more important at Rayleigh numbers and viscosity contrasts more representative of the mantles of terrestrial planets. The robustness of the N𝓊-Ra relation, when scaled by Racrit, is important for studies of planetary thermal history that rely on parameterizations of convective heat transport and account for temperature dependence of mantle viscosity.

Oscillatory two- and three-dimensional thermocapillary convection

The character and stability of two- and three-dimensional thermocapillary driven convection are investigated by numerical simulations. In two dimensions, Hopf bifurcation neutral curves are delineated for fluids with Prandtl numbers ( Pr ) 10.0, 6.78, 4.4 and 1.0 in the Reynolds number ( Re )–cavity aspect ratio ( A x ) plane corresponding to Re [les ]1.3×10 4 and A x [les ]7.0. It is found that time-dependent motion is only possible if A x exceeds a critical value, A xcr , which increases with decreasing Pr . There are two coexisting neutral curves for Pr [ges ]4.4. Streamline and isotherm patterns are presented at different Re and A x corresponding to stationary and oscillatory states. Energy analyses of oscillatory flows are performed in the neighbourhood of critical points to determine the mechanisms leading to instability. Results are provided for flows near both critical points of the first unstable region with A x =3.0 and Pr =10. In three dimensions, attention is focused on the influence of sidewalls, located at y =0 and y = A y , and spanwise motion on the transition. In general, sidewalls have a damping effect on oscillations and hence increase the magnitude of the first critical Re . However, the existence of spanwise waves can reduce this critical Re . At large aspect ratios A x = A y =15, our results with Pr =13.9 at the lower critical Reynolds number of the first unstable region are in good agreement with those from infinite layer linear stability analysis.

Infinite Prandtl number thermal convection in a spherical shell

A Galerkin method is used to calculate the finite amplitude, steady, axisymmetric convective motions of an infinite Prandtl number, Boussinesq fluid in a spherical shell. Convection is driven by a temperature difference imposed across the stress-free, isothermal boundaries of the shell. The radial gravitational field is spherically symmetric and the local acceleration of gravity is directly proportional to radial position in the shell. Only the case of a shell whose outer radius is twice its inner radius is considered. Two distinct classes of axisymmetric steady states are possible. The temperature and radial velocity fields of solutions we refer to as ‘even’ are symmetric about an equatorial plane, while the latitudinal velocity is antisymmetric about this plane; solutions we refer to as ‘general’ do not possess any symmetry properties about the equatorial plane. The characteristics of these solutions, i.e. the isotherms, streamlines, spherically averaged temperature profiles, Nusselt numbers, etc., are given for Rayleigh numbers Ra as high as about 10 times critical for the even solutions and 3 times critical for the general solutions. Linear stability analyses of the nonlinear steady states show that the general solutions are the preferred form of axisymmetric convection when Ra is less than about 4 times critical. Furthermore, while the preferred motion at the onset of convection is non-axisymmetric, axisymmetric convection is stable when Ra exceeds about 1·3 times the critical value.

Character and stability of axisymmetric thermal convection in spheres and spherical shells

Abstract Nonlinear axisymmetric convective motions of self-gravitating, infinite Prandtl number fluids in spheres and thick spherical shells are determined for a variety of shell sizes and for different modes of heating. For one combination of heating from within and from below the onset of convection is governed by a self-adjoint system of equations and boundary conditions. For two other heating modes, heating only from within or only from below, the linearized equations and boundary conditions are not self-adjoint. The properties of the self-adjoint solutions together with a parameter which quantifies the departure from self-adjointness provide a theoretical framework for organizing, understanding, and generalizing the heat transfer characteristics of the non-self-adjoint cases. The variations in heating and shell size that are considered yield 6 different patterns of steady convection—flows with 1 and 3 meridional cells, and pairs of oppositely rotating flows with 2 and 4 meridional cells. The heat tra...

Absolute and convective instability of a cylinder wake

Local linear stability theory is used to investigate the stability characteristics of a circular cylinder wake at values of the Reynolds number, Re, up to 45. Both the Orr–Sommerfeld and Rayleigh analyses for complex frequency and wavenumber are applied to computed basic, steady wake profiles. These velocity profiles are Navier–Stokes solutions of a uniform, incompressible viscous flow around a cylinder, which we obtain by a spectral method. The distributions of convective and absolute instabilities in the near wake behind the cylinder are determined for different Reynolds numbers and locations. The study shows that an absolutely unstable region begins to form at Re of about 20, and grows with increasing Re. Hence the onset of global response of instability, which is known to occur at Re of about 40, must be characterized by a critical length of an absolutely unstable region. A criterion to determine the critical Reynolds number Rec and the preferred frequency, from local linear stability analysis, is pro...