Sinusoidal Boundary Condition Research Articles

On finite element method–flux corrected transport stabilization for advection–diffusion problems in a partial differential–algebraic framework

An extension of the finite element method–flux corrected transport stabilization for hyperbolic problems in the context of partial differential–algebraic equations is proposed. Given a local extremum diminishing property of the spatial discretization, the positivity preservation of the one-step θ-scheme when applied to the time integration of the resulting differential–algebraic equation is shown, under a mild restriction on the time step size. As a crucial tool in the analysis, the Drazin inverse and the corresponding Drazin ordinary differential equation are explicitly derived. Numerical results are presented for non-constant and time-dependent boundary conditions in one space dimension and for a two-dimensional advection problem with a sinusoidal inflow boundary condition and the advection proceeding skew to the mesh.

Effect of aspect ratio on natural convection in an inclined rectangular enclosure with sinusoidal boundary condition

The aim of the present numerical study is to understand the natural convection flow and heat transfer in an inclined rectangular enclosure with sinusoidal temperature profile on the left wall. The top and bottom walls of the enclosure are kept to be adiabatic. The finite difference method is used to solve the governing equations with a range of inclination angles, aspect ratios and Rayleigh numbers. The results are presented in the form of streamlines, isotherms and Nusselt numbers. The heat transfer increases first then decreases with increasing the inclination of the enclosure for all aspect ratio and Rayleigh number. Increasing the aspect ratio shows a decreasing trend of the heat transfer for all Rayleigh numbers considered. A correlation equation is also introduced for the heat transfer analysis in this study.

Influence of periodicity of sinusoidal bottom boundary condition on natural convection in porous enclosure

In this paper, natural convective flow within a rectangular enclosure has been investigated numerically. All the walls of the enclosure are adiabatic except the bottom wall, which is partially heated and cooled by sinusoidal temperature profile. Both situations: medium is hydro-dynamically isotropic and anisotropic are considered. The governing equations are written under assumption of Brinkman-extended non-Darcy model, including material derivative, and then solved by numerically using spectral element method (SEM). Main emphasize is given on effect of periodicity parameter (N) on local heat transfer rate (Nux) as well as flow mechanism in the enclosure. The result shows that, as the periodicity is decreased on increasing N, the absolute value of Nux at the bottom left corner point increases in both situations. For odd values of N, the local heat transfer profile is symmetric about the line X=0.5. The entire flow is governed by two type convective cells: (i) rotating clockwise (ii) rotating anticlockwise. Furthermore, increasing of N increases the multiple cellular structures in the form of N+1. The relative impact of media permeability on Nux at (0,0) shows that enhancement of media permeability along Y-direction is much more effective than the same along X-direction. In contrast to isotropic porous media, the effect of the permeability and permeability orientation angle on the flow is complex and often non intuitive. In particular the present analysis shows that, different periodicity of temperature boundary condition has significant effect on the flow mechanism and consequently on the heat transfer rate for both situations.

Influence of Non-Uniform Sinusoidal Periodic Bottom Boundary Condition on Natural Convection in an Isotropic Porous Enclosure

A comprehensive numerical investigation on the natural convection in an isotropic porous enclosure is presented. All the walls of the enclosure are adiabatic except the bottom wall which is partially heated and cooled by sinusoidal temperature profile. The governing equations were written under assumption of Brinkman-extended non-Darcy model, including material derivative, and then solved by numerically using spectral element method (SEM). The heat transfer and fluid flow mechanisms in isotropic case are governed by periodicity parameter (N) Rayleigh Number (Ra), Darcy number (Da), aspect ratio (A), Prandtl number (Pr) and media permeability (K). The main emphasize is given on effect of N on local heat transfer as well as mechanism of heat transfer and fluid flow in enclosure. The results shows that, as the periodicity is decreased on increasing N the absolute value of Nux at the bottom left corner point increases. For odd values of N, the local heat transfer profile is symmetric about the line x=0.5, which is consequence of symmetric boundary condition at the bottom wall of the enclosure. The entire flow is governed by two type convective cells: (i) rotating clockwise (ii) rotating anticlockwise. Furthermore for even values of N cells rotating anticlockwise are dominated and covered the entire domain. In particular the present analysis shows that, different periodicity of temperature boundary condition has the significant effect on the flow mechanism and consequently on the heat transfer rate.

Numerical prediction of turbulent flow structure generated by a synthetic cross‐jet into a turbulent boundary layer

SUMMARYA detailed numerical study using large‐eddy simulation (LES) and unsteady Reynolds‐averaged Navier–Stokes (URANS) was undertaken to investigate physical processes that are engendered in the injection of a circular synthetic (zero‐net mass flux) jet in a zero pressure gradient turbulent boundary layer. A complementary study was carried out and was verified by comparisons with the available experimental data that were obtained at corresponding conditions with the aim of achieving an improved understanding of fluid dynamics of the studied processes. The computations were conducted by OpenFOAM C++, and the physical realism of the incoming turbulent boundary layer was secured by employing random field generation algorithm. The cavity was computed with a sinusoidal transpiration boundary condition on its floor. The results from URANS computation and LES were compared and described qualitatively and quantitatively. There is a particular interest for acquiring the turbulent structures from the present numerical data. The numerical methods can capture vortical structures including a hairpin (primary) vortex and secondary structures. However, the present computations confirmed that URANS and LES are capable of predicting current flow field with a more detailed structure presented by LES data as expected. Copyright © 2011 John Wiley & Sons, Ltd.

Thermal dynamic transfer properties of the opaque envelope: Analytical and numerical tools for the assessment of the response to summer outdoor conditions

The reliable estimation of buildings energy needs for cooling is a crucial issue in the implementation of the EPB Directive 2010/31/EU (formerly 2002/91/EC), especially in central and southern Europe climates. On this purpose one of the main topics is to predict the behavior of the opaque envelope subjected to variable boundary conditions. The EN ISO 13786:2007 technical standard assumes sinusoidal boundary conditions and introduces dynamic thermal characteristics. The aim of this paper is to assess the deviation arising by the use of different approaches for the calculation of the dynamic thermal characteristics of an opaque envelope element under periodic non sinusoidal boundary conditions. The EN ISO 13786 procedure has been firstly applied by decomposing the external forcing temperature by means of the Fast Fourier Transform (FFT) analysis. A comparison with different approaches, such as Finite Difference Methods (FDM) and Transfer Function Methods (TFM), has been carried out. The predictions of the EN ISO 13786 with a sinusoidal forcing temperature (i.e., standard approach) have also been assessed, comparing the results to the ones obtained through the FFT analysis. Furthermore, corrections to the periodic thermal transmittance and to the time shift have been proposed, in order to improve the explicative worth of those parameters.

Hydro-magnetic combined convection in a lid-driven cavity with sinusoidal boundary conditions on both sidewalls

Mixed convection in a square cavity of sinusoidal boundary temperatures at the sidewalls in the presence of magnetic field is investigated numerically. The horizontal walls of the cavity are adiabatic. The governing equations are solved by finite-volume method. The results are obtained for various combinations of amplitude ratio, phase deviation, Richardson number, and Hartmann number. The heat transfer rate increases with the phase deviation up to ϕ = π/2 and then it decreases for further increase in the phase deviation. The heat transfer rate increases on increasing the amplitude ratio. The flow behavior and heat transfer rate inside the cavity are strongly affected by the presence of the magnetic field.

Assessment of mixing problem on the EOF with thermal effects

Abstract A potential of mixing applications on an electroosmotic flow (EOF) with thermal effects is examined. For the thermal conditions, we apply the sinusoidal temperature boundary conditions on the walls. We exemplify two cases: (1) the mixing of laminar flows and (2) Taylor–Aris dispersion model. In the first case, we consider to mix two different samples that flow in parallel along the channel. In addition, by scaling analysis, we qualitatively examined the mixing result. The mixing efficiency is proportional to the temperature difference. Through the Taylor–Aris dispersion model, we found that the temperature gives rise to an increase of the D eff( T) at the low Peclet number where the diffusion and convection effect coexist.

Assessment of near-surface ground temperature profiles for optimal placement of a thermoelectric device

For a thermoelectric device driven by the temperature difference between soil and ambient air, previous analytical work has been performed wherein a sinusoidal surface boundary condition was imposed; the results suggested that optimal placement of the lower terminal of the device should be at a nondimensional depth of x ∗ = 2.28, resulting in a 7% increase in power over placement at a much greater depth. Analysis of temperature data taken in conjunction with operation of a thermoelectric device has shown that the optimal depth for the lower thermal reservoir is much shallower than originally thought ( x ∗ ∼ 1), with increases in performance approaching 70% over that experienced at greater depths. Representative data are presented and interpreted, along with recommendations for future work.

Combined convection in inclined porous lid-driven enclosures with sinusoidal thermal boundary condition on one wall

A numerical study has been performed to obtain combined convection field in an inclined porous lid-driven enclosures heated from one wall with a non-uniformly heater. The lid moves at constant speed and temperature. Darcy-Brinkman-Forchheimer model was adopted to write governing equations and they solved using finite volume method. A detailed parametric study has been presented for different Reynolds numbers (10 ≤ Re ≤ 1000), Grashof numbers (104 ≤ Gr ≤ 105), Darcy numbers (0.01 ≤ Da ≤ 0.1), porosity (0.6 ≤ ε ≤ 0.9) and inclination angle of enclosure (0° ≤ φ ≤ 180°). It is observed that flow field, temperature distribution and heat transfer are affected by inclination angle of the enclosure.

A moving boundary problem derived from heat and water transfer processes in frozen and thawed soils and its numerical simulation

The seasonal change in depths of the frozen and thawed soils within their active layer is reduced to a moving boundary problem, which describes the dynamics of the total ice content using an independent mass balance equation and treats the soil frost/thaw depths as moving (sharp) interfaces governed by some Stefan-type moving boundary conditions, and hence simultaneously describes the liquid water and solid ice states as well as the positions of the frost/thaw depths in soil. An adaptive mesh method for the moving boundary problem is adopted to solve the relevant equations and to determine frost/thaw depths, water content and temperature distribution. A series of sensitivity experiments by the numerical model under the periodic sinusoidal upper boundary condition for temperature are conducted to validate the model, and to investigate the effects of the model soil thickness, ground surface temperature, annual amplitude of ground surface temperature and thermal conductivity on frost/thaw depths and soil temperature. The simulated frost/thaw depths by the model with a periodical change of the upper boundary condition have the same period as that of the upper boundary condition, which shows that it can simulate the frost/thaw depths reasonably for a periodical forcing.

Effects of discrete-electrode configuration on traveling-wave electrohydrodynamic pumping

Traveling-wave electrohydrodynamic (EHD) micropumps can be incorporated into the package of an integrated circuit chip to provide active cooling. They can also be used for fluid delivery in microdevices. The pump operates in the presence of a thermal gradient through the fluid layer such that a gradient in electrical conductivity is established allowing ions to be induced. These ions are driven by a traveling electric field. Such a traveling electric field can be realized in practice only via discrete electrodes upon which the required voltages are imposed. The impact of using discrete electrodes to create the traveling wave on the flow rates generated is explored through numerical modeling. The change in performance from an ideal sinusoidal voltage boundary condition is quantified. The model is used to explore the widths of electrodes and the intervening isolation regions that lead to optimized pumping. The influence of the choice of working fluid on the performance of the pump is determined using an analytical model.

Experimental investigation on a mechanically pumped two-phase cooling loop with dual-evaporator

This paper describes a well-known space activate thermal control system, two-phase mechanically pumped cooling loop (MPCL) with two evaporators, in ground-based testing. Each evaporator has an outer diameter of 3 mm and a length of 10 m and the total loop of the system is about 40 m. Tests conducted mainly included stable operation tests under sinusoidal temperature boundary condition and unbalanced heat load tests. Experimental data showed that the loop operated stably at about 30 °C variation of the boundary temperature without any temperature overshoot. Meanwhile, the operating temperature could be controlled within ±0.1 °C of the set point temperature with the new design based on TECs. Furthermore, heat load unbalanced between the two evaporators was also successfully demonstrated that the loop has a heat transport capability to deal with the maximal unbalanced heat loads limitation as Top_Load/Bottom_Load = 0 W/160 W or Top_Load/Bottom_Load = 130 W/0 W. When the heat load limitation is reached in one evaporator and a dry-out appears, some heat load employed on the other side such as 10 W will recover it to the set point evaporation temperature. In conclusion, important performance characteristics demonstrated by this MPCL included: (1) operation of the system about 30 m with a mechanical pump; (2) robustness and reliability of an MPCL with multiple evaporators; (3) effectiveness of TECs in controlling the MPCL operating temperature; (4) dealing with a large unbalanced heat load among MPCL evaporators; and (5) easy recovery from the pulse boiling or a dried out state.

Growth/decay of transverse acceleration waves in nonlinear elastic media

A study of transverse acceleration wave phenomena in the context of nonlinear elasticity is carried out. Using singular surface theory, exact amplitude expressions for acceleration waves arising from a sinusoidal boundary condition (BC) are obtained and their temporal evolution and propagation speeds determined. It is shown that amplitude blow up is possible only if the medium ahead is in motion, otherwise a constant amplitude wave occurs. Analytical results are illustrated numerically using simple computational methods/tools. Lastly, erroneous results that were reported in a recently published related study are noted and/or corrected.

Optimal placement depth for air–ground heat transfer systems

Heat transfer between the air and the ground is important for a variety of applications including buried pipes and cables, ground source heat pumps, and building insulation. A recent application considered a ground-source heat engine using a thermoelectric module sandwiched between two heat exchangers to produce a very small amount of electricity from the daily temperature difference between the air and the ground. For a given device, the power produced is proportional to the square of the temperature difference across the module and it is desirable to maximize the power. For a sinusoidal (in time) temperature boundary condition, the solution to the heat equation for a semi-infinite body yields an attenuated sinusoid shifted in phase from the boundary condition. The phase shift and attenuation both increase with depth, so while the phase shift may be exploited to increase the temperature difference, the attenuation tends to reduce this effect. In this paper, an optimal depth for the ground side heat exchanger is derived which maximizes the square of the difference between the surface temperature and the ground temperature. The maximum temperature difference in the optimized case is 7% higher than what would be obtainable from an infinitely deep heat exchanger. The range of depths in which the temperature difference is an improvement over the infinitely deep case is derived and presented. The improvement is asymmetric so that away from the optimum depth, it declines rapidly with decreasing depth while it declines slowly with increasing depth. An alternative optimization that reflects the temperature drop through the heat engine device is derived and presented. The optimal depth decreases by slightly more than half in this case. The effect of a convection boundary condition at the free surface is explored in terms of a Biot number for the ground. The Biot number affects both the attenuation and the phase shift. The attenuation is approximately doubled for Biot of order unity, and becomes inversely proportional to Biot number if it is much less than unity. The phase shift varies from an additional shift of 45° for Biot much less than unity to an additional shift of 27° for Biot of order unity. For Biot much greater than unity, the temperature boundary condition applies.

Periodic forcing in composite aquifers

Observations of periodic components of measured heads have long been used to estimate aquifer diffusivities. The estimations are often made using well-known solutions of linear differential equations for the propagation of sinusoidal boundary fluctuations through homogeneous one-dimensional aquifers. Recent field data has indicated several instances where the homogeneous aquifer solutions give inconsistent estimates of aquifer diffusivity from measurements of tidal lag and attenuation. This paper presents new algebraic solutions for tidal propagation in spatially heterogeneous one-dimensional aquifers. By building on existing solutions for homogeneous aquifers, comprehensive solutions are presented for composite aquifers comprising of arbitrary (finite) numbers of contiguous homogeneous sub-aquifers and subject to sinusoidal linear boundary conditions. Both Cartesian and radial coordinate systems are considered. Properties of the solutions, including rapid phase shifting and attenuation effects, are discussed and their practical relevance noted. Consequent modal dispersive effects on tidal waveforms are also examined via tidal constituent analysis. It is demonstrated that, for multi-constituent tidal forcings, measured peak heights of head oscillations can seem to increase, and phase lags seem to decrease, with distance from the forcing boundary unless constituents are separated and considered in isolation.

Application of Differing Forcing Function Models on Simulated Flow Past an Oscillating Cylinder in a Uniform Low Reynolds Number Flow

A Computational Fluid Dynamics (CFD) model is presented for the uniform viscous two dimensional flow past an oscillating cylinder at low Reynolds number. Numerical simulations are made to study the effect of differing forced induced oscillation mechanisms with a large range of cylinder forcing frequencies. In the first case sinusoidal velocity slip boundary conditions are adopted for the cylinder surface to simulate cylinder oscillation. The implication suggests that no modification or additional term need to be added to the Navier-Stokes equations. In the second case this time extra body force terms which are assumed to account for velocity effects due to cylinder movement are included in the Navier-Stokes equations with the imposition of same boundary conditions. Drag and lift coefficients are extracted from present numerical results and other detailed computations of these coefficients are made at a Reynolds number of 80 and an amplitude-to diameter ratio 0.14. The results are found to be in agreement with each other at low force driving frequencies below and near lock-in. However, differences are found at higher frequencies above lock-in. Agreement are also found with experimental results at some frequency ranges.

Analytical Solutions for Partitioned Diffusion in Laminates: II. Harmonic Forcing Conditions

The migration of organic compounds in stratified media is of fundamental concern in environmental and chemical engineering research. The diffusive transport of volatile organic compounds through laminate systems is characterized by partitioning, i.e., the development of concentration discontinuities at the interfaces between the individual laminae. If the transport is governed by cyclic transients, the relevant equations can be written in terms of coupled systems of diffusion equations subject to sinusoidal boundary conditions. This paper solves these systems of equations to present new algebraic solutions for propagation of the sinusoidal modes through arbitrary (finite) numbers of contiguous one-dimensional laminae. Both Cartesian and radial coordinate systems are considered. Two independent formulations of the lamina interface matching conditions are considered, corresponding to (1) an instantaneous partitioning model and to (2) a mass-limited partitioning model. It is shown that sinusoidal components of the concentration solutions propagate dispersively throughout the laminates. This is manifested, for example, in changes in shape of concentration pulses as measured at different points in the laminate system. Algorithms for generating exact dispersion relations for partitioning laminates are given, and an experimental technique for studying interfacial dynamics via frequency domain measurements is proposed.

Separation of gas mixtures in unsteady-state conditions

The prospects for the application of unsteady-state boundary conditions at the membrane inlet for increasing the selectivity of gas separation are discussed in this paper. The phenomena occurring upon passage of a concentration pulse and two-gas-component penetrant concentration waves through the membrane have been investigated. It has been shown that pulsed supply of the mixture to be separated at the membrane inlet increases the separation coefficients by a factor of several orders owing to differences in the diffusion coefficients of the gas mixture components in the membrane. Sinusoidal boundary conditions at the membrane inlet allow filtration of the amplitude of the total output oscillations from the signal of the component with a low diffusion coefficient (in this case the membrane acts as a frequency filter), which can be employed for increasing the selectivity of the sensors. The proposed techniques are exemplified by separation of the He/CO 2 gas mixture on a polymeric polyvinyltrimethylsilane (PVTMS) membrane.

Calculation of magnetic domain wall bowing and eddy current losses in an infinite sheet of bar-like domains

An analytic solution has been obtained for the magnetic field distribution and eddy current losses associated with an infinite array of regularly spaced 180° domain walls which are allowed to bow during their motion. Wall motion was constrained to satisfy a sinusoidal total flux boundary condition. The field solution was solved self-consistently with the domain wall equation of motion. Calculated losses were always lowered relative to the plane wall case. Difficulties were encountered in the numerical iteration scheme as the degree of bowing increased. Using an approximation for the high bowing case, a physical instability occurred before bowing became too severe.