Rigid Lower Boundary Research Articles

A Variational Method for Analyzing Vortex Flows in Radar-Scanned Tornadic Mesocyclones. Part I: Formulations and Theoretical Considerations

AbstractA variational method is formulated with theoretical considerations for analyzing vortex flows in Doppler radar–scanned tornadic mesocyclones. The method has the following features. (i) The vortex center axis (estimated as a continuous function of time and height in the four-dimensional space) is used as the vertical coordinate, so the coordinate system used for the analysis is slantwise curvilinear and nonorthogonal in general. (ii) The vortex flow (VF), defined by the three-dimensional vector wind minus the horizontal moving velocity of vortex center axis, is expressed in terms of the covariant basis vectors (tangent to the coordinate curves), so its axisymmetric part can be properly defined in that slantwise-curvilinear coordinate system. (iii) To satisfy the mass continuity automatically, the axisymmetric part is expressed by the scalar fields of azimuthally averaged tangential velocity and cylindrical streamfunction and the remaining asymmetric part is expressed by the scalar fields of streamfunction and vertically integrated velocity potential. (iv) VF-dependent covariance functions are formulated for these scalar variables and then deconvoluted to construct the square root of background error covariance matrix analytically with the latter used to transform the control vector to precondition the cost function. (v) The deconvoluted covariance functions and their transformed control variables satisfy two required boundary conditions (i.e., zero vertical velocity at the lower rigid boundary and zero cross-axis velocity along the vortex center axis), so the analyzed VF satisfies not only the mass continuity but also the two boundary conditions automatically.

Behaviour of circular footings confined by rigid base and geocell reinforcement

Soil in nature may exist in a stratified state. A possible condition is the case of a rigid layer underlying a weak sand stratum. In this instance, the bearing capacity of a shallow footing is affected by the lower rigid boundary. The treatment of thin soil layer by geocell reinforcement to raise the load carrying capacity of single shallow footing has been limited in the literature. Accordingly, rigid base and geocell reinforcement are investigated separately and in combination to study their influence on the behaviour and BC of shallow circular footings in dry sand bed. To capture that effect, different size rigid circular footings were tested on a bed in fully instrumented small- and large-scale laboratory installations. The ratio of sand layer thickness to footing diameter was changed considering optimum dimension and embedment depth for geocell mattress. The extensive laboratory part involved monitoring subgrade deformations, soil-geocell stresses and strains revealing the appropriate models for failure mechanisms in the presence of such confinements. Large increase in bearing capacity up to 225% and significant settlement reduction up to 66% are measured for the combined case. Base confinement less than three times the footing diameter in combination with the geocell mattress resulted in favourable higher design performance factors. New equations are proposed to estimate these factors for design and to extend classical formulations for footings strengthened by combined geocell-base confinement. A design performance factor is dependent on footing diameter, sand relative density and the ratio of layer thickness to footing diameter. As this ratio increases, this design factor decreases reaching a constant value at the critical depth.

A numerical study on interface shearing of granular Cosserat materials

This article presents the interface shearing behaviour of a granular soil layer in contact with a bounding structure under motion. For this purpose, an elasto-plastic Cosserat continuum model (EPCCM) and finite element method (FEM) in the updated Lagrange frame are employed. Plane quasi-static shearing of a granular soil layer, bounded by upper and lower rigid boundaries with different surface roughness conditions, is simulated under free dilatancy and constant vertical pressure. Herein, the additional Cosserat kinematical boundary constraints along the bounding surfaces allow a more detailed description of the surface roughness. Specific emphasis is given to the effects of surface roughness as well as the boundary conditions of the entire system on the shear behaviour of layer. According to the obtained results, the shear band location is different in an infinite or finite shear layer depending on the adopted boundary constraints. Further numerical simulations of plane interface shearing are also carried out here using discrete element method (DEM), complementary to Cosserat FE analyses, to better understand the micro-mechanics of granular media near interfaces. FE results are compared with those of experiments as well as DEM simulations and reasonable agreements are seen. The evolution and distribution of state variables and polar quantities in DEM simulations are in proper accordance with the predictions of Cosserat FE model.

Finite-amplitude perturbations of the advective flows in a horizontal layer of incompressible liquid with a free upper boundary at low rotation

The behavior of spatially finite-amplitude perturbations of the advective flow in a rotating horizontal layer of incompressible liquid with a free upper boundary and a lower rigid boundary at low rotation is investigated. The study is based on the equations of convection in the Boussinesq approximation in a rotating reference frame in a Cartesian coordinate system. Because of the great complexity of a three-dimensional formulation the following limiting cases are considered: spatial perturbations of the first type in the form of rolls with axes perpendicular to x -direction and spatial perturbations of the second type in the form of rolls with axes parallel to x -axis. At a fixed Prandtl number (Pr=6.7), perturbation isolines are plotted for different Taylor and Grashof numbers. The behavior of finite-amplitude perturbations is considered beyond the stability threshold. The nonlinear problem is solved numerically by the grid method. An explicit finite-difference scheme with central differences is used. The Poisson equation for perturbation stream functions is solved by the successive over-relaxation (SOR) method. The numerical results allow one to evaluate the behavior of perturbations and to determine velocity characteristics, amplitude and period of repetition of perturbations. The analysis indicates that spatial structures (vortices) oriented across the layer are generated in the supercritical region under the influence of temperature inhomogeneities. Temperature perturbation is a system of alternating warm and cold spots located along the direction of the temperature gradient on the layer boundaries. With increasing Taylor number, the vortices near the free upper boundary are replaced by those along the lower rigid boundary, and the amplitude of vortices increases. With increasing Grashof number, thermal spots are rearranged, their size increases, the warm spots are located near the upper and lower boundaries of the layer, and the cold ones in the center of the layer. The movement becomes more complex. A repetition period of finite amplitude perturbation patterns reduces.

Effect of temperature-dependent viscosity on the onset of Bénard–Marangoni ferroconvection in a ferrofluid saturated porous layer

The criterion for the onset of Bénard–Marangoni ferroconvection in an initially quiescent magnetized ferrofluid saturated horizontal Brinkman porous layer is investigated in the presence of a uniform vertical magnetic field. The viscosity is considered to be varying exponentially with temperature. The lower rigid boundary and the upper free boundary at which the surface tension effects are accounted for are assumed to be perfectly insulated to temperature perturbations. The eigenvalue problem is solved numerically using the Galerkin technique and analytically by regular perturbation technique with wave number a as a perturbation parameter. It is observed that the analytical and numerical results are very well comparable. The characteristics of stability of the system are strongly dependent on the viscosity parameter B. The effect of B on the onset of Bénard–Marangoni ferroconvection in a porous layer is dual in nature depending on the choices of the physical parameters, and a sublayer starts to form at higher values of B. The nonlinearity of fluid magnetization M 3 is found to have no influence on the onset of ferroconvection, whereas an increase in the value of the magnetic number M 1 and the Darcy number Da is to advance the onset of Bénard–Marangoni ferroconvection in a porous layer.

Dynamical model of faulting in two dimensions and self-healing of large fractures

We describe a model for the simulation of extended two-dimensional in-plane dynamical ruptures and for the rapid calculation of statistical properties of repeated model-seismicity events. The discretization involves first- and second-nearest neighbors and is isotropic in both compression and shear properties. All rupture events obey a fracture criterion in the appropriate coordinate frame and numerical oscillations in slip velocity at crack tips due to discretization are minimized. The rupture velocities of fractures, in cases of homogeneous stress drop equal to the strength, are the supershear P-wave velocity in the direction of the prestress and the S-wave velocity in the perpendicular direction. We use the model to study the growth and healing of individual faults to understand the formation of propagating slip pulses. We confirm two mechanisms for the generation of isolated rupture pulses that have been proposed, namely, (1) a decrease in the dynamical friction with accelerating slip and (2) the encounter of the growing crack with extended regions of large difference between the threshold fracture stress and the prestress. We describe a third mechanism which is that of a velocity-dependent friction that operates equally on both the phases of increasing and decreasing slip velocities and has a characteristic length scale. It is a proxy for energy loss by radiation in a three-dimensional medium. In the case of an elongated rectangular model fault with an upper free surface and lower rigid boundary, pulses develop due to the influence of stress waves reflected from the rigid bottom boundary. In general, the excess of strength over stress drop controls crack fracture speeds; if it is too large, the crack stops. Under homogeneous stress conditions, isolated slip pulses are controlled by the spatial distribution of heterogeneities and by the velocity-dependent friction parametrization.

Synthetic Aperture Radar Remote Sensing of Shear-Driven Atmospheric Internal Gravity Waves in the Vicinity of a Warm Front

Abstract The synthetic aperture radar ocean surface signature of atmospheric internal gravity waves in the vicinity of a synoptic-scale warm front is examined via a classic Kelvin–Helmholtz velocity profile with a rigid lower boundary and a sloping interface. The horizontal distance that the waves extend from the surface warm front is consistent with a bifurcation along the warm frontal inversion from unstable to neutral solutions. Similarity theories are derived for the wave span and the location of maximum growth rate relative to the surface front position. The theoretical maximum wave growth rate is demonstrated to occur near this bifurcation point and, hence, to explain the observed pattern of wave amplitude. Finally, a wave crest-tracing procedure is developed to explain the observed acute orientation of waves with respect to the surface warm front.

Bio-thermal convection induced by two different species of microorganisms

This paper develops a theory of bio-thermal convection in a suspension that contains two species of microorganisms exhibiting different taxes, gyrotactic and oxytactic microorganisms. The developed theory is applied to investigating the onset of bio-thermal convection in such a suspension occupying a horizontal layer of finite depth. A linear stability analysis is utilized to derive the equations for the amplitudes of disturbances. The obtained eigenvalue problem is solved by the Galerkin method. The case of non-oscillatory instability in a layer with a rigid lower boundary and a stress-free upper boundary is investigated. The resulting eigenvalue equation relates three Rayleigh numbers, the traditional Rayleigh number (Ra) and two bioconvection Rayleigh numbers, one for gyrotactic (Rb g) and one for oxytactic (Rb o) microorganisms. The neutral stability boundary is presented in the form of a diagram showing that boundary in the (Ra, Ra g) plane for different values of Ra o.

Brinkman–Benard–Marangoni convection in a magnetized ferrofluid saturated porous layer

A classical linear stability theory is used to study the onset of Brinkman–Benard–Marangoni (BBM) convection in an initially quiescent magnetized ferrofluid saturated horizontal layer of a very coarse porous medium in the presence of a uniform vertical magnetic field. The lower rigid boundary is subject to a fixed heat flux, while the upper boundary is free and open to the ambient air on which a general thermal condition which encompasses fixed temperature and fixed heat flux as particular cases is invoked. The resulting eigenvalue problem is solved using the Galerkin technique and also by a regular perturbation technique when both boundaries are insulated to temperature perturbations. The differences as well as similarities with the corresponding problem in ordinary viscous fluids and also the influence of different forces when they are acting in isolation are particularly highlighted. It is found that increase in the Biot number, porous parameter and decrease in the magnetic number as well as nonlinearity of fluid magnetization has a stabilizing effect on the onset of BBM ferroconvection. Moreover, the nonlinearity of fluid magnetization is observed to have no consequence on the onset of convection in the case of fixed heat flux boundary conditions. The present study reproduces already established results in the literature as particular cases.

Long-wavelength Marangoni Convection in a Thermally Modulated Field

This paper is concerned with the long-wave Marangoni instability in a horizontal liquid layer. The analysis has been carried out for the heat flux periodically oscillating at the deformable upper interface and at the rigid lower boundary. This type of instability is caused by the amplification of synchronous disturbances in response to the external forcing. Convection thresholds are determined, and the critical Marangoni number is expressed as a function of frequency. It is shown that the long-wavelength mode can give rise to the instability prior to the cellular mode within the definite range of parameters. The long-wavelength instability can have an influence on the space-qualified manufacturing, as well as the industrial process of thin films deposition on substrates.

The comparative analysis of two solutions of Pekeris boundary problem

The solution of the reduced Pekeris boundary problem, satisfying to the generalized radiation condition on an impedance interface is obtained. Mode part of the solution is presented by series expansion on the total system of regular normal waves, of generalized normal waves and of leakage normal waves. The principle of the continuous continuation of the obtained solution in physical half‐space with validity of boundedness condition is formulated. Expansion of the solution on physical half‐space in a class of generalized functions is constructed. The obtained generalized solution includes canonical series expansion on the set of eigenfunctions of both conjugate operators corresponding an initial not self‐conjugate boundary problem, the associated eigenfunctions of a waveguide with two soft boundaries, the associated eigenfunctions of a waveguide with upper soft boundary and lower rigid boundary and a set of spherical components. Results of the comparative analysis of the classical solution and generalized solution are given.

A Baroclinic Instability that Couples Balanced Motions and Gravity Waves

Abstract Normal modes of a linear vertical shear (Eady shear) are studied within the linearized primitive equations for a rotating stratified fluid above a rigid lower boundary. The authors' interest is in modes having an inertial critical layer present at some height within the flow. Below this layer, the solutions can be closely approximated by balanced edge waves obtained through an asymptotic expansion in Rossby number. Above, the solutions behave as gravity waves. Hence these modes are an example of a spatial coupling of balanced motions to gravity waves. The amplitude of the gravity waves relative to the balanced part of the solutions is obtained analytically and numerically as a function of parameters. It is shown that the waves are exponentially small in Rossby number. Moreover, their amplitude depends in a nontrivial way on the meridional wavenumber. For modes having a radiating upper boundary condition, the meridional wavenumber for which the gravity wave amplitude is maximal occurs when the tilts of the balanced edge wave and gravity waves agree.

Upper bounds on convective heat transport in a rotating fluid layer of infinite prandtl number: case of intermediate taylor numbers

By means of the Howard-Busse method of the optimum theory of turbulence we obtain upper bounds on the convective heat transport in a heated from below layer of fluid of infinite Prandtl number rotating with a constant angular velocity about the vertical axis. We consider the region of intermediate Taylor numbers: alpha41<<Ta<<alpha61 where alpha(1) is the wave number connected to the 1-alpha-solution of the variational problem. The studied optimum fields possess a three-layer or four-layer structure: in addition to the internal, intermediate, and boundary layers, Ekman layers could arise between the intermediate and boundary ones. For the discussed interval of Taylor numbers the intermediate layers do not expand in the direction of the internal layers. We present an asymptotic theory for the case of the fluid layer with rigid lower boundary and stress-free upper boundary. We use an improved solution of the Euler-Lagrange equations of the variational problem for the intermediate sublayer of the optimum field. This solution leads also to correction of the thicknesses of the boundary layers and to lowering of the upper bounds on the convective heat transport for the cases of fluid layer with stress-free or with rigid boundaries. Thus the known upper bounds for these cases can be treated as upper bounds on the upper bounds on the convective heat transport. For the case of the fluid layer with stress-free boundaries the four-layer optimum fields leads to bounds on the convective heat transport which change from R(1/3) at the lower boundary of their interval of validity to values slightly large than R(2/7) near the upper boundary of the interval of validity. Finally we discuss the area of application of the obtained bounds with respect to the Taylor number Ta and Rayleigh number R.

Upper bound on the heat transport in a layer of fluid of infinite prandtl number, rigid lower boundary, and stress-free upper boundary

We obtain an upper bound on the convective heat transport in a heated from below horizontal fluid layer of infinite Prandtl number with rigid lower boundary and stress-free upper boundary. Because of the asymmetric boundary conditions the solutions of the Euler-Lagrange equations of the corresponding variational problem are also asymmetric with different thicknesses of the boundary layers on the upper and lower boundary of the fluid. The obtained bound on the convective heat transport and the corresponding wave number are between the values for a fluid layer with two rigid boundaries and a fluid layer with two stress-free boundaries.

The effect of a uniform magnetic field on the onset of steady Marangoni convection in a layer of conducting fluid with a prescribed heat flux at its lower boundary

In this paper a combination of analytical and numerical techniques are used to analyze the effect of a uniform vertical magnetic field on the onset of steady Marangoni convection in a horizontal layer of quiescent, electrically conducting fluid with a uniform vertical temperature gradient subject to a prescribed heat flux at its rigid lower boundary. Critical Marangoni numbers for the onset of instability are calculated which are significantly different from those calculated previously in the case of an isothermal lower boundary. Analytical results for the behavior of the critical Marangoni number in the asymptotic limit of large magnetic field strength are also obtained. It is concluded that the magnetic field always has a stabilizing effect on the onset of steady Marangoni convection, but that when the free surface is deformable situations with a sufficiently large Marangoni number will always have unstable modes no matter how strong the applied magnetic field is.

Analytic solutions for soil‐structure interaction in layered media

AbstractThe total system studied in this paper is a layered soil stratum with a rigid bedrock and a cylindrical cavity on the surface. Analytic solutions for the layered medium with prescribed harmonic displacement time history on the surface of the cylindrical cavity are presented. The whole soil domain is divided into interior and exterior domains. The interior domain is the projection of the cylindrical cavity down to the rigid bedrock, whereas the exterior domain is then the soil medium complement to the interior domain. The displacement and stress fields in both domains are expanded as an infinite series of Fourier components with respect to the azimuth. For each Fourier component in the infinite series, the solutions for both domains are found independently by solving the general differential equations of wave propagation satisfying the boundary conditions of the top surface and the lower rigid boundary. Displacement and stress continuity conditions are then imposed on the vertical interface between the two domains using the formulation of a weighted residual. For the soil‐structure interaction problem, the impedance matrix at the interface between the structure and the soil medium can be easily generated using the analytic solutions, which can then be combined with the finite element model of the structure. A simple example is presented to demonstrate the effectiveness of the procedure presented.

The propagation of two-dimensional and axisymmetric viscous gravity currents at a fluid interface

Viscous gravity currents resulting from the introduction of fluid between an upper layer of fluid of lesser density and a lower layer of greater density are analysed. The nonlinear equations governing the spread and shape of the intrusion are formulated for the cases of intrusion at low Reynolds number between deep ambient layers and of flow over a shallow layer of viscous fluid with a rigid lower boundary. Similarity solutions of these equations are obtained in both two-dimensional and axisymmetric geometries, under the assumption that the volume of intruding fluid increases with time like tα. The theoretical predictions are shown to be in reasonable agreement with experimental observations of the spreading of glucose syrups and of viscous hydrocarbons between fluid layers of differing densities. Scaling arguments are used to derive many new results for the rates of spread of intrusions in a wide variety of further situations. A compendium of spreading relations, including some previously isolated results, is derived within a coherent framework and tabulated.

Transient thermal convection in a rotating porous medium confined between two rigid boundaries

The onset of steady and transient thermal convection in a rotating porous medium, bounded by upper and lower rigid boundaries, is considered. The initial temperature distribution is linear and is increased from the below at a constant heating rate. The global effects of both friction and form drags, based on the Darcy's law, are described by the macroscopic governing equations, using the volume averaged techniques. Linear stability theory predicts the critical Rayleigh number and wavenumber, marking the marginal state. The transient thermal convection is treated as an initial value problem. Linear amplification theory is applied to determine the critical time of the onset of transient thermal convection.

Shear Instabilities in the Atmosphere in the Presence of a Jump in the Brunt-Väisälä Frequency

Abstract The stability properties of a Helmholtz velocity profile in a stratified, Boussinesq fluid are investigated in the presence of a jump in the Brunt- Vaisala (BV) frequency at a level different from the one where the vortex sheet is located. New unstable modes in the range of low horizontal wavenumbers are found with respect to the case where no BV frequency jump exists. The structure of the associated unstable disturbances is similar to that of neutrally propagating gravity waves. The associated growth rates are rather small but significant because they appear in a range of horizontal wavenumbers which are otherwise stable. A comparison with the results obtained by Lindzen and Rosenthal and by Lalas et al. shows a strict analogy with the study of the stability properties of a Helmholtz velocity profile in the presence of a lower rigid boundary. The generation of such instabilities is interpreted in terms of multiple overreflexions at the shear interface due to the presence of the reflecting BV fre...

Response of an isothermal bounded atmosphere to an applied random body-force

The response of a one dimensional isothermal atmosphere with rigid lower boundary and bounded upward by a very hot isothermal atmosphere to an applied random source is studied. It is shown that a concentrated white noise source merely excites the damped eigen-modes of the model and that the enhanced response at the cut-off frequency found in semi-infinite isothermal model is removed by the reflections. We believe that this rules out the attempts to relate the photospheric and chromospheric oscillations to the singular response of an excited semi-infinite isothermal atmosphere.