Sidewall Boundary Conditions Research Articles

Electrothermal CAD of power devices and circuits with fully physical time-dependent compact thermal modeling of complex nonlinear 3-d systems

An original, fully analytical, spectral domain decomposition approach is presented for the time-dependent thermal modeling of complex nonlinear (3-D) electronic systems, from metallized power FETs and MMICs, through MCMs, up to circuit board level. This solution method offers a powerful alternative to conventional numerical thermal simulation techniques, and is constructed to be compatible with explicitly coupled electrothermal device and circuit simulation on CAD timescales. In contrast to semianalytical, frequency space, Fourier solutions involving DFT-FFT, the method presented here is based on explicit, fully analytical, double Fourier series expressions for thermal subsystem solutions in Laplace transform s-space (complex frequency space). It is presented in the form of analytically exact thermal impedance matrix expressions for thermal subsystems. These include double Fourier series solutions for rectangular multilayers, which are an order of magnitude faster to evaluate than existing semi-analytical Fourier solutions based on DFT-FFT. They also include double Fourier series solutions for the case of arbitrarily distributed volume heat sources and sinks, constructed without the use of Green's function techniques, and for rectangular volumes with prescribed fluxes on all faces, removing the adiabatic sidewall boundary condition. This combination allows treatment of arbitrarily inhomogeneous complex geometries, and provides a description of thermal material nonlinearities as well as inclusion of position varying and non linear surface fluxes. It provides a fully physical, and near exact, generalized multiport network parameter description of nonlinear, distributed thermal subsystems, in both the time and frequency domains. In contrast to existing circuit level approaches, it requires no explicit lumped element, RC-network approximation or nodal reduction, for fully coupled, electrothermal CAD. This thermal impedance matrix approach immediately gives rise to minimal boundary condition independent compact models for thermal systems. Implementation of the time-dependent thermal model as N-port netlist elements within a microwave circuit simulation engine, Transim (NCSU), is described. Electrothermal transient, single-tone, two-tone, and multitone harmonic balance simulations are presented for a MESFET amplifier. This thermal model is validated experimentally by thermal imaging of a passive grid array representative of one form of spatial power combining architecture.

A three-dimensional, two-way, parabolic equation model for acoustic backscattering in a cylindrical coordinate system

A new PE model for solving three-dimensional, forward and backward sound propagation in a cylindrical coordinate system is presented. The model marches a wave field in the radial direction including the azimuthal diffraction effects, and solves for a backscattered field based on a three-dimensional, single scattering approach. A periodic sidewall boundary condition is applied for computations in a 360-degree sector, while an approximate sidewall boundary condition is used for calculation in a sector less than 360 degrees. These two sidewall boundary conditions are verified by the numerical results. The major drawback of using the cylindrical coordinate system, when the backscattering solution is valid within a limited area, is analyzed using a geometrical-optical interpretation. The model may be useful for studying three-dimensional backscattering phenomena comprising azimuthal diffraction effects.

Natural convection in a rectangular cavity heated from below and cooled from top as well as the sides

A three-dimensional, numerical simulation of Rayleigh–Benard convection in a cavity, having aspect ratios Ax=4 and Ay=4 has been performed using multigrid method, for Ra⩽1×104. The effect of the thermal boundary conditions of the vertical side walls on the flow bifurcations has been investigated by considering isothermal and conducting side wall cases. It has been found that the steady state flow patterns and the critical Rayleigh numbers for transition from one pattern to the other are completely different for each of the above mentioned boundary conditions. The local heat transfer distributions at the bottom and top plates are obtained and variations of the average Nusselt number for the isothermal and conducting side wall boundary conditions are compared for the range of Rayleigh numbers investigated.

Multiplicity of equilibrium states in laterally heated thermosolutal systems with equal diffusivity coefficients

This work uncovers the instabilities arising in laterally heated stably stratified systems when the diffusivities of the two involved components are equal. These instabilities are demonstrated to be the result of the differential diffusion caused by the unequal lateral diffusion gradients of the components. Such gradients form in the perturbed state due to the different side-wall boundary conditions. Examination of the bifurcation phenomena in the finite enclosures with equal diffusivities exhibited most qualitative features established by Tsitverblit and Kit [Phys. Fluids A 5, 1062 (1993)] and Tsitverblit [Phys. Fluids 7, 718 (1995)] for such phenomena in the heat-salt problem. In the behavior of singularities and steady flows, a number of the regularities that are not distinct in the heat-salt case were distinguished. Additional results obtained with the solute sidewall boundary conditions being of the same (fixed-value) type as the temperature conditions were also discussed.

Three-dimensional eddy structure in a cylindrical container

We consider Stokes flow in a cylindrical container of circular section induced by the uniform translatory motion of one of the endwalls. This flow field is of interest because it is possible to get reliable analytical descriptions of important three-dimensional structures such as the primary and corner eddies. It is shown, using a result of Tran-Cong & Blake, that separable solutions exist which can be combined to yield vector eigenfunctions that satisfy the sidewall boundary conditions provided the eigenvalues satisfy the transcendental equationformula here

Mode Interactions in Large Aspect Ratio Convection

Mode interaction between odd and even modes in two-dimensional Boussinesq convection in a box is revisited. It is noted that in the large aspect ratio limit the structure of the amplitude equations depends on the boundary conditions applied at the sidewalls, however distant.With no-slip sidewall boundary conditions the equations approach those for an unbounded layer with periodic boundary conditions; this is not the case for free-slip boundary conditions. Thus only in the former case can the large aspect ratio system be considered a small perturbation of the unbounded system. The reasons for the different large aspect ratio limits are traced to the presence of “hidden” symmetries in the stress-free case. Homotopic continuation is used to extend these results to other types of boundary conditions.

Asymptotic analysis of a vertical Bridgman furnace at large Rayleigh number

A vertical Bridgman furnace, through which an ampoule containing the melt of a certain dilute binary alloy is pulled at a fixed, and predetermined speed, provides a means of improving certain alloys. Radial segregation in the finished crystalline material can make it unusable. In this paper, we construct an asymptotic theory for the flow and solidification in the ampoule, for large Rayleigh numbers but at a small Biot number. We find large-Rayleigh-number solutions to be, to leading order, completely insensitive to the character of the side-wall boundary condition on vertical velocity. Two-dimensional equivalents of optimization conditions found by Tanveer are recovered for his two limiting cases—large thermal Rayleigh number, and large negative solutal Rayleigh number. Moderate surface tension at the crystal–melt interface is found to have no effect on the optimization conditions for the two limit cases, but it does somewhat reduce the magnitude of the segregation in both limits. In addition, we present new results for the case for which the two Rayleigh numbers are of comparable magnitude and show that there is an optimization possible for this case too. Conversion of the results of this paper to an axisymmetric geometry is shown to be trivial. Keeping careful track of the ordering, we indicate how to proceed to first effects of non-linearity at small Biot and/or Prandtl number.

Observations of Spanwise Symmetry Breaking for Unsteady Mixed Convection in Horizontal Ducts

Three-dimensional mixed convection in a horizontal rectangular duct heated from below was studied numerically for a non-Boussinesq fluid. The duct was of aspect ratio four, with cold side-wall boundary conditions. Results were obtained at a reduced temperature of 2.33, Rayleigh number of 130,700, and Reynolds numbers of 5 and 20. Using the Re = 20 solution as initial conditions, the Re = 5 solution developed asymmetries about the duct centerline plane, with a streamwise periodic generation and dissolution of the inner pair of vortices. The consequence of these coherent vortical structures was a complex time-dependent distribution of the Nusselt number on the lower wall, with implications in the area of chemical vapor deposition. Specifically, when compared with the steady flow case occurring at Re = 20, an increased spatial uniformity of the time-averaged Nusselt number along the lower duct surface resulted.

Interfacial and Buoyancy-Driven Convection—The Effect of Geometry and Comparison with Experiments

In this study, pattern formation at the bifurcation point from the quiescent state for free surface convection in circular containers is determined. A linearized instability calculation that employs three-dimensional disturbances in the presence of physically realistic side wall boundary conditions is made. The results of the present study will provide a very useful asymptotic limit for nonlinear numerical computations. Under restrictive conditions the current calculations check favorably with those of earlier workers. The results of these studies are compared with the experiments of Koschmieder and Prahl (J. Fluid Mech.251,571, 1990).

Study of a model of thermal convection in cylindrical containers

We propose a generalized Swift-Hohenberg model equation that shows different planforms as observed in experiments in convective layers heated from below. We use this model to study the natural planform when some symmetric inhomogeneities are present on the sidewall boundary conditions. A stability analysis in the 1D case allow us to determine when the rolls parallel to the sidewalls remain stable or perturbations perpendicular to these sidewalls can appear. Having this simple analysis in mind we study numerical solutions of the model equation in 2D considering those inhomogeneities. This shows that, even near threshold and for sufficiently high inhomogeneities, a pattern of concentric rolls can be a stable solution. However, it is replaced by a pattern with a different symmetry when these inhomogeneities decay or are suppressed. This is in qualitative agreement with experimental observations.

Hopf bifurcations in Langmuir circulations

The equations governing two-dimensional Langmuir circulations in a continuously stratified layer of fluid possess O(2) symmetry when laterally periodic boundary conditions are applied. In one limit, which is among those treated here, the mathematical problem is strictly analogous to a double-diffusive problem. Steady, oscillatory and multiple oscillatory states are all possible. The method of multiple scales is used to obtain evalution equations for the amplitudes of oscillatory convection when a single wavenumber is destabilized, and when two wavenumbers are simultaneously destabilized. In the process, this paper provides the first example of the derivation of amplitude equations for a double-Hopf bifurcation with symmetry in a fluid mechanical problem. Parallel numerical simulations of the full partial differential equations are carried out, and quantitative comparisons are made between the two methods, both when periodic boundary conditions are enforced, and when the more restrictive flux-free side-wall boundary conditions are enforced.

Computation of convective flow with gravity modulation in rectangular cavities

In this work, a computational study is presented for the investigation of gravity modulation (g-jitter) effects in thermally driven cavity flows at terrestrial and microgravity environments. The two-dimensional, time-dependent Navier-Stokes equations are numerically integrated by a time-split method using direct matrix solvers. Computations at terrestrial gravity are utilized to assess the effects of adiabatic side-wall boundary conditions as well as the full nonlinearity of the governing equations on the sinusoidally forced Benard problem studied by Gresho and Sani. The low-g calculations focus on the establishment of critical frequency ranges and consider the effects of modulation direction and randomness. The applicability of linear analysis in the excitable frequency range at low g is also discussed.

Three-Dimensional Laminar Natural Convection in a Vertical Air Slot With Hexagonal Honeycomb Core

Numerical solutions are obtained for a three-dimensional natural convection heat transfer problem in a vertical air slot with a thin hexagonal honeycomb core. The air slot is assumed to be of such dimensions that the velocity and temperature fields repeat themselves in successive enclosures. The numerical methodology is based on an algebraic coordinate transformation technique, which maps the complex cross section onto a rectangle, coupled with a calculation procedure for fully elliptic three-dimensional flows. The calculations are performed for the Rayleigh number in the range of 103 to 105, for a Prandtl number of 0.7, and for five values of the aspect ratio of the honeycomb enclosure. The average Nusselt number results for the case of a thin honeycomb core are compared with the previously obtained results for a thick honeycomb core with conduction and adiabatic side wall boundary conditions.

Study on the Effects of Side-Wall Interference on Added-Mass and Damping Coefficients by Means of Slender-Ship Theory

Slender-ship theory is applied to develop a new method of calculating with quantitatively good accuracy the effects of side-wall interference on the added-mass and damping coefficients of a ship in a waterway. The present paper is restricted to the radiation problem of heave and pitch motions with zero forward speed, but it may be easily possible to extend to the forward-speed case and the diffraction problem.In the outer region, 3-D Green function satisfying the boundary condition on the side walls is placed along the ship's centerline on the free surface to describe the outer solution. The side-wall boundary condition is satisfied by considering an infinite number of image ships, and then the summation of the resultant infinite series is analytically obtained in a closed form.In the inner region, the radiation condition and side walls are absent, and thus the inner solution is identical to that of Newman's unified theory : the strip-theory solution plus a homogeneous solution multiplied by a three-dimensional interaction coefficient. This interaction coefficient is determined by the compatibility requirement of the inner and outer solutions in an overlap region. Therefore through this coefficient, the side-wall effects are incorporated in the inner solution.Computations of the added-mass and damping coefficients are performed for a floating spheroid of length-beam ratio L/B=8 in a channel of breadth ratio B T/B=16, and of L/B in a channel of B T/B = 5. Numerical results are in virtually perfect agreement with the results by a newly-developed 3-D panel method. This suggests that the theory proposed can be quite effectively used also in estimating the tank-wall effects on the results of experiments in a towing tank with limited width.

Steady, Two-Dimensional, Natural Convection in Rectangular Enclosures With Differently Heated Walls

Numerical results are presented for steady natural convection in two-dimensional rectangular enclosures in which the side walls, top wall, and bottom wall are at uniform temperatures θs, θt, and θb, respectively, and θs > θt > θb. Raylight numbers ranging from 104 to 107 and aspect ratios of 1 and 1.5 were investigated. The top wall was modeled as an impermeable rigid surface or an impermeable free-moving boundary. The calculations reveal two flow regions. In the upper part of the enclosure two large counterrotating cells appear, separated by a descending plume of fluid. Near the bottom wall the flow is almost motionless and stably stratified. The temperature in the central portion of the enclosure is almost uniform due to mixing by the recirculating cells. A temperature inversion occurs near the top wall and is particularly noticeable at high Rayleigh numbers. At high Rayleigh numbers the flow breaks up into smaller cells. The result is that each main recirculation region develops a secondary counterrotating eddy within it. The condition of a free surface as the top wall boundary condition significantly affects the circulation and heat transfer throughout the flow domain. Numerical experiments reveal the extent to which the flow field in the enclosure is affected by an asymmetric specification of side-wall temperature boundary conditions.

The effect of a continuously deforming coastline on the numerical simulation of storm surges in Bangladesh

In the present paper a vertically integrated coastal zone numerical model has been described for the simulation of storm surges and associated coastal inundation in Bangladesh. The model is a generalization of the earlier study [3] in which the fixed side-wall lateral boundary condition along the coast of Bangladesh has been replaced by a continuously deforming fluid boundary and includes the coastal topography in order to route the storm surges over the land. Numerical experiments are performed with the help of this model in order to simulate the surge generated by the November 1970 Chittagong cyclone. A comparison is made between simulations using models both with and without inland intrusion of water along the Bangladesh coast.

Confinement of thermocapillary floating zone flow by uniform rotation

The microgravity environment of an orbiting vehicle permits crystal growth experiments with greatly reduced buoyant convection in the liquid melt. Crystals grown in ground-based laboratories do not achieve their potential properties because of dopant variations caused by flow in the melt. The floating zone crystal growing system is widely used to produce crystals of silicon and other materials. However, in this system the temperature gradient on the free sidewall surface of the melt is the source of a thermocapillary flow which does not disappear in a low gravity environment. Smith and Greenspan examined theoretically the idea of using a uniform rotation of the floating zone system to confine the thermocapillary flow to the melt sidewall leaving the interior of the melt passive. These workers considered a half zone, a cylinder of fluid with a constant axial temperature gradient imposed on the cylindrical sidewall. They examined the linearized, axisymmetric flow in the absence of crystal growth. They found that rotation does confine the linear thermocapillary flow. In this paper the simplified model of Smith and Greenspan is extended to a full zone with a more realistic temperature distribution imposed on the sidewall and both linear and nonlinear thermocapillary flows are studied theoretically. Analytical and numerical methods are used for the linear flows and numerical methods for the nonlinear flows. We found that the linear flows in the full zone have more complicated and thicker boundary layer structures than in the half zone, and that these flows are also confined by the rotation. For the model considered and for realistic values for silicon, however, the thermocapillary flow is not linear. We examined the nonlinear flows by first computing a weakly nonlinear flow and then computing the fully nonlinear flow. The weakly nonlinear flow is steady, has less boundary layer character, and penetrates more deeply into the interior than the linear flow but still shows some rotational confinement. The fully nonlinear flow is strong and unsteady (a weak oscillation is present) and it penetrates the interior. Some non-rotating flow results are also presented. Since silicon has a large value of thermal conductivity, we would expect the temperature fields to be determined by conduction alone. This is true for the linear and weakly nonlinear results, but for the stronger nonlinear flow the results show that temperature advection is also important. Thus, this work reveals that for the nonlinear flow, a radiative sidewall boundary condition would be an improvement over the specified temperature boundary condition used in this paper and previously by others. Such a boundary condition would weaken the sidewall axial temperature gradient and hence the thermocapillary flow, allowing the confining effect of rotation to play a stronger role. Hence, uniform rotation may still be a means of confining the flow, and the results obtained define the procedure to be used to examine this hypothesis.

Flow Structure with Natural Convection in Inclined Air-Filled Enclosures

Flow visualization observations are described for natural convection flow in rectangular inclined enclosures. Observations are made in air-filled enclosures of small and moderate aspect ratio (0.25 ⩽ Ax ⩽ 7), angles of inclination from 0 to 90 deg and Rayleigh number between 5 × 103 and 2.5 × 105. For the range of parameters considered we determined the transition from stationary to nonstationary flow and the transition from two-dimensional to three-dimensional flow. Also, the different stationary flow structures are described. In addition measurements of the velocity profile of the stationary flow have been performed by means of a laser doppler anemometer. Special attention has been given to the side wall boundary condition of the enclosure.

Coastal boundary layers in ocean modelling: an application to the Adriatic Sea

Boundary layers play an important role in modelling geophysical fluid-dynamical flows, in as much as they constitute regions of ageostrophic dynamics in which the physical balances characterizing the main interior of the water mass break down. A short synopsis is given of important boundary layers in ocean circulation modelling with specific emphasis drawn upon side wall boundary layers, namely those adjacent to the coastlines of the considered basin. Application of boundary layer analysis is thereafter made for one specific phenomenological situation, namely the Northern Adriatic Sea and the problem posed by its wintertime seasonal circulation. The analysis furnishes a mathematical model for the coastal strip adjacent to the Italian shoreline, treated as a boundary layer in the density field, starting from general model equations valid throughout the interior of the northern Adriatic. The boundary layer model is consequently used to modify the side wall boundary condition for the interior density field. Related numerical experiments are shown and compared with previous standard experiments in which the boundary layer contribution to the density field has not been considered.

Application of line-edge profile simulation to thin-film deposition processes

A computer simulation program based on a segment-motion model has been developed to investigate thin-film-deposition processes. This simulation program is used to study the physical phenomena in the thin-film-deposition process as well as the effects of different tooling configurations and operating conditions. The simulated results are compared with experimental results obtained from an electron-gun evaporator with planetary tooling and from a planarizing sputtered quartz system to verify the accuracy of the simulation and to illustrate imporant phenomena during the deposition process. The simulation program is also used in exploratory studies to find the optimum source angle for step coverage in a typical planetary evaporator. The step coverage is found to change only slightly for wafers at different locations on the planet, but can change significantly depending on the pattern orientation on the wafers if the wafer rotation is not independent of the planet rotation. A study of the sputtered quartz planarization process shows that the growth of different topological features can be accurately predicted from the change in deposition and etching rate as a function of the angle of incidence. In particular, the increased deposition near a conductor edge is attributed to the propagation of a growth front from the sidewall boundary conditions.