Bottom Boundary Condition Research Articles

Design safe distance between a buried pipe and a soil slope

Leakage from pipes buried in a soil slope can lead to failure and scour of the slope. Placing the pipe at a safe distance from the slope crest is a key concern in design. The objectives of the work reported here were to investigate the wetting front advancement in a soil slope due to leakage from buried pipes at variable distances from the slope crest considering different soil densities and boundary conditions of the slope, and to recommend a safe separation distance based on the relation between the factor of safety of the slope and the separation distance. Stability charts for slopes were established for different separation distances and infiltration times, which will serve as a guide to determining a safe separation distance. The safe distance depends on soil permeability, slope geometry and infiltration time. A safe distance, regardless of how long the leakage is expected to last, is recommended as twice the slope height for soil slopes 15 m in height and 35° in inclination compacted to 95% relative compaction under impermeable side and bottom boundary conditions. When a pipe is laid within this distance, the ability to detect and repair pipe leakage in a timely manner becomes an essential means of maintaining slope stability.

Effect of bottom surface optical boundary conditions on nanofluid-based DASC: Parametric study and optimization

In this study, low-flux direct absorption solar collectors (DASCs) with nanofluid volume absorbers were modeled, analyzed, and optimized. The Rayleigh scattering approximation with size-dependent effects was used in order to determine nanofluid optical properties. Upon validating the mathematical model resulting from numerically solving and coupling the energy conservation equation with the radiative transfer equation, effects of internal bottom-surface optical boundary condition and base-fluid type on nanofluid temperature homogeneity and collector first- and second-law efficiencies were studied for different particle loadings, film thicknesses, and nanoparticle materials. Non-linear multi-variable constrained single- and multi-objective global optimization studies were conducted to find the optimal design vectors with respect to first- and/or second-law objective functions. The type of bottom surface in a DASC was shown to significantly affect its performance, particularly for relatively low particle loadings. Beyond a critical nanoparticle volume fraction value, collector performance was independent of bottom surface type and a DASC operates similar to a surface absorber. It has been also found that regardless of particle loading and bottom surface type, collectors with thinner nanofluid films always had a lower efficiency compared to collectors with thicker films. Water-based nanofluids were shown to offer stronger radiation absorption than therminol®-based ones up to a nanoparticle volume fraction of about 0.005%, at which level, therminol® becomes the stronger solar absorber. However, it was established that a nanofluid exhibiting stronger photo-thermal conversion does not necessarily lead to a higher collector efficiency. Finally, it was shown that optimizing with respect to a normalized combination of energy and exergy efficiencies (as opposed to only energy or exergy efficiencies) results in more reasonable design vectors with a balance between collector power and temperature gains.

Effects of the bottom boundary condition in numerical investigations of dense water cascading on a slope

The flow of dense water along continental slopes is considered. There is a large literature on the topic based on observations and laboratory experiments. In addition, there are many analytical and numerical studies of dense water flows. In particular, there is a sequence of numerical investigations using the dynamics of overflow mixing and entrainment (DOME) setup. In these papers, the sensitivity of the solutions to numerical parameters such as grid size and numerical viscosity coefficients and to the choices of methods and models is investigated. In earlier DOME studies, three different bottom boundary conditions and a range of vertical grid sizes are applied. In other parts of the literature on numerical studies of oceanic gravity currents, there are statements that appear to contradict choices made on bottom boundary conditions in some of the DOME papers. In the present study, we therefore address the effects of the bottom boundary condition and vertical resolution in numerical investigations of dense water cascading on a slope. The main finding of the present paper is that it is feasible to capture the bottom Ekman layer dynamics adequately and cost efficiently by using a terrain-following model system using a quadratic drag law with a drag coefficient computed to give near-bottom velocity profiles in agreement with the logarithmic law of the wall. Many studies of dense water flows are performed with a quadratic bottom drag law and a constant drag coefficient. It is shown that when using this bottom boundary condition, Ekman drainage will not be adequately represented. In other studies of gravity flow, a no-slip bottom boundary condition is applied. With no-slip and a very fine resolution near the seabed, the solutions are essentially equal to the solutions obtained with a quadratic drag law and a drag coefficient computed to produce velocity profiles matching the logarithmic law of the wall. However, with coarser resolution near the seabed, there may be a substantial artificial blocking effect when using no-slip.

Updating the Coupling Algorithm between HYDRUS and MODFLOW in the HYDRUS Package for MODFLOW

Core Ideas The coupling algorithm between HYDRUS‐1D and MODFLOW is updated. Pressure head profiles are updated based on the groundwater table and bottom flux after each time step. This eliminates sudden variations in inflow and outflow fluxes with groundwater table changes. The HYDRUS‐based flow package for MODFLOW (the HPM or the HYDRUS package) is an existing unsaturated zone flow package for MODFLOW. In MODFLOW with the HPM, the groundwater modeling domain is discretized into regular grids that can be combined into multiple zones based on similarities in soil hydrology, topographical characteristics, and the depth to the groundwater. Each of these zones is assigned one unsaturated soil profile (the HPM profile). In this model, after every MODFLOW time step, the flux at the bottom of the HPM profile is given as an input recharge flux to MODFLOW. MODFLOW simulates groundwater flow, and the water table depth at the end of the MODFLOW time step is assigned as the bottom boundary condition in the HPM profile. The current coupling algorithm assumes that the groundwater table in the HPM profile remains constant throughout the entire MODFLOW time step. This results in unrealistic sudden inflow and/or outflow fluxes at the bottom of the HPM profile after every time step. The objective of this study was to develop a methodology to eliminate the error in the determination of the recharge flux at the bottom of the HPM profile. This was achieved by updating or modifying the pressure head profile in the HPM profile after every MODFLOW time step. The effectiveness and the applicability of the new coupling algorithm were evaluated using different case studies. The new coupling algorithm is effective in eliminating unrealistic sudden variations in the bottom flux in the HPM profiles.

The effect of mixed convection on the thermal field of horizontal channel flow

An experimental study was conducted to investigate the thermal field of low-speed flow of moistair in a horizontal rectangular duct, uniformly heated from below (Poiseuille–Benard flow). Averaged temperature fields were measured in vertical mid-plane using thermocouples. The buoyancy-induced secondary flow is interacting with forced Poiseuille flow which leads to complex flow behaviour with flow reversal along the top surface. Two types of bottom boundary conditions were examined: aluminium plate (only heat transfer) and water film (simultaneous heat and mass transfer) heated to the same temperature. Presence of mass transfer causes stronger transient behaviour above the bottom surface and enhancing of heat transfer, but the overall character of the thermal field remains the same as in case of heat transfer only.

The effect of bottom boundary condition type on the behavior of adhesive contact of spherical probe on an elastic film

This study presents an investigation on the behavior of adhesive contact between a rigid sphere and an elastic film which is either perfectly bonded (case I) or in frictionless contact (case II) with a rigid substrate. By using linear fracture mechanics, we formulate an convenient semi-analytical approach to develop relations between the applied force, penetration depth and contact radius. Finite element analysis (FEA) is used to verify the relationships. Our results reveal that the interfacial boundary conditions between the film and substrate have distinct effects on the adhesive contact behavior between the sphere and the film. The aim of the present study is to provide an instructive inspiration for controlling adhesion strength of the thin film subject to adhesive contact.

Modelling hydraulic jump using the bubbly two-phase flow method

Hydraulic jumps have complex flow structures, characterised by strong turbulence and large air contents. It is difficult to numerically predict the flows. It is necessary to bolster the existing computer models to emphasise the gas phase in hydraulic jumps, and avoid the pitfall of treating the phenomenon as a single-phase water flow. This paper aims to improve predictions of hydraulic jumps as bubbly two-phase flow. We allow for airflow above the free surface and air mass entrained across it. We use the Reynolds-averaged Navier–Stokes equations to describe fluid motion, the volume of fluid method to track the interface, and the k–e model for turbulence closure. A shear layer is shown to form between the bottom jet flow and the upper recirculation flow. The key to success in predicting the jet flow lies in formulating appropriate bottom boundary conditions. The majority of entrained air bubbles are advected downstream through the shear layer. Predictions of the recirculation region’s length and air volume fraction within the layer are validated by available measurements. The predictions show a linear growth of the shear layer. There is strong turbulence at the impingement, and the bulk of the turbulence kinetic energy is advected to the recirculation region via the shear layer. The predicted bottom-shear-stress distribution, with a peak value upstream of the toe of the jump and a decaying trend downstream, is realistic. This paper reveals a significant transient bottom shear stress associated with temporal fluctuations of mainly flow velocity in the jump. The prediction method discussed is useful for modelling hydraulic jumps and advancing the understanding of the complex flow phenomenon.

Quasi-3D two-phase model for dam-break flow over movable bed based on a non-hydrostatic depth-integrated model with a dynamic rough wall law

The assumptions of equilibrium flow conditions neglecting the flow acceleration for vertical velocity distributions and of the equilibrium wall law applied to the bottom boundary condition, in which the flow acceleration is neglected in the vicinity of the bed, are questionable when used to calculate sediment transport for flows that vary rapidly over time and space. Examples of such flows include dam-break flows and flows around structures. This study proposes a two-phase depth-integrated model for large-scale geophysical flow applications involving sediment transport phenomena and bed morphology. The model for the fluid phase is based on the non-hydrostatic quasi-3D method, and uses a dynamic rough wall law that employs continuity and momentum equations for the bottom boundary conditions.It was confirmed that the proposed model reduces to the previous bedload formulae for uniform flow conditions under the weak sediment transport condition. The model was applied in an experiment involving a dam-break flow on a movable bed channel with a suddenly enlarged section. The comparison between the experimental results and the results calculated with the proposed model and previous models demonstrates the validity of the proposed model. The comparison also highlights the advantages of introducing a quasi-3D two-phase model to evaluate vertical velocity distributions and non-equilibrium sediment motions.

A theoretical model of the variation of the meridional circulation with the solar cycle

Observations of the meridional circulation of the Sun, which plays a key role in the operation of the solar dynamo, indicate that its speed varies with the solar cycle, becoming faster during the solar minima and slower during the solar maxima. To explain this variation of the meridional circulation with the solar cycle, we construct a theoretical model by coupling the equation of the meridional circulation (the $\phi$ component of the vorticity equation within the solar convection zone) with the equations of the flux transport dynamo model. We consider the back reaction due to the Lorentz force of the dynamo-generated magnetic fields and study the perturbations produced in the meridional circulation due to it. This enables us to model the variations of the meridional circulation without developing a full theory of the meridional circulation itself. We obtain results which reproduce the observational data of solar cycle variations of the meridional circulation reasonably well. We get the best results on assuming the turbulent viscosity acting on the velocity field to be comparable to the magnetic diffusivity (i.e. on assuming the magnetic Prandtl number to be close to unity). We have to assume an appropriate bottom boundary condition to ensure that the Lorentz force cannot drive a flow in the subadiabatic layers below the bottom of the tachocline. Our results are sensitive to this bottom boundary condition. We also suggest a hypothesis how the observed inward flow towards the active regions may be produced.

Конвективный теплообмен вязкой несжимаемой жидкости в цилиндрической ячейке с конически углубленным дном и твердыми граничными условиями

Конвективный теплообмен вязкой несжимаемой жидкости в цилиндрической ячейке с конически углубленным дном и твердыми граничными условиями

Possible management of near shore nonlinear surging waves through bottom boundary conditions

We propose an alternative way for managing near shore surging waves, including extreme waves like tsunamis, going beyond the conventional passive measures like the warning system. We study theoretically the possibility of influencing the nonlinear surface waves through a leakage boundary effect at the bottom. It has been found through analytic result, that the controlled leakage at the bottom might regulate the amplitude of the surface solitary waves. This could lead to a possible decay of the surging waves to reduce its hazardous effects near the shore. Our theoretical results are estimated by applying it to a real coastal bathymetry of the Bay of Bengal in India.

Modeling glacial flow on and onto Pluto’s Sputnik Planitia

Observations of Pluto’s surface made by the New Horizons spacecraft indicate present-day N2 ice glaciation in and around the basin informally known as Sputnik Planitia. Motivated by these observations, we have developed an evolutionary glacial flow model of solid N2 ice that takes into account its published thermophysical and rheological properties. This model assumes that glacial ice flows laminarly and has a low aspect ratio which permits a vertically integrated mathematical formulation. We assess the conditions for the validity of laminar N2 ice motion by revisiting the problem of the onset of solid-state buoyant convection of N2 ice for a variety of bottom thermal boundary conditions. Subject to uncertainties in N2 ice rheology, N2 ice layers are estimated to flow laminarly for thicknesses less than 400–1000 m. The resulting mass-flux formulation for when the N2 ice flows as a laminar dry glacier is characterized by an Arrhenius–Glen functional form. The flow model developed is used here to qualitatively answer some questions motivated by features we interpret to be a result of glacial flow found on Sputnik Planitia. We find that the wavy transverse dark features found along the northern shoreline of Sputnik Planitia may be a transitory imprint of shallow topography just beneath the ice surface suggesting the possibility that a major shoreward flow event happened relatively recently, within the last few hundred years. Model results also support the interpretation that the prominent darkened features resembling flow lobes observed along the eastern shoreline of the Sputnik Planitia basin may be the result of a basally wet N2 glacier flowing into the basin from the pitted highlands of eastern Tombaugh Regio.

Analysis of the Bearing Capacity of Shallow Foundation in Unsaturated Soil Using Monte Carlo Simulation

The ultimate bearing capacity of shallow foundation supported by unsaturated soil depends on the degree of saturation of the soil within the influence zone because the strength and deformation parameters of soil are affected by the degree of saturation. As the degree of saturation varies with rainfall, surface runoff, evapotranspiration and other climatic and geotechnical parameters, these parameters must be systematically incorporated for accurately computing the ultimate bearing capacity. In this study, a framework is proposed to compute the ultimate bearing capacity of a shallow footing in unsaturated soil considering site specific rainfall and water table depth distributions. The randomness in rainfall and water table depth is systematically considered using Monte Carlo method. The infiltration of water through the unsaturated zone is modelled using Richards equation considering infiltration and water table location as the top and bottom boundary conditions, respectively. The results show that the bearing capacity calculated using the proposed method is approximately 2.7 times higher than that calculated using the deterministic approach with fully saturated soil parameters.

Experimental and Numerical Study of the Onset of Transient Natural Convection in a Fractured Porous Medium

In this study, analysis of transient natural convection in a porous medium with a vertical fracture across it in a rectangular enclosure was performed experimentally and numerically. A number of thermocouples in distinctive distances in the model were used to measure temperature distribution and confirm the numerical simulations. A detailed CFD simulation was carried out, which was based on the finite volume method to extend our experimental observations. The onset of transient natural convection in the fractured porous media was observed by experimental setup and successfully simulated by numerical analysis. A modified Rayleigh number for the constant surface temperature at the bottom boundary condition of the model was calculated in this study (i.e., use of $$K_\mathrm{avg}$$ instead of K), which was near to the theoretical value of linear stability analysis. Moreover, the critical time of the onset convection in the fractured model was predicted. The average Nusselt number was estimated to be about 3.5 for this fractured porous model.

Froude scaling limitations in modeling of turbidity currents

The scaling problem associated with the modeling of turbidity currents has been recognized but is yet to be explored systematically. This paper presents an analysis of the dimensionless governing equations of turbidity currents to investigate the scale effect. Three types of flow conditions are considered: (i) conservative density current; (ii) purely depositional turbidity current; and (iii) mixed erosional/depositional turbidity current. Two controlling dimensionless numbers, the Froude number and the Reynolds number, appear in the non-dimensional governing equations. When densimetric Froude similarity is satisfied, the analysis shows that the results would be scale-invariant for conservative density current under the rough turbulent condition. In the case of purely depositional flows, truly scale-invariant results cannot be obtained, as the Reynolds-mediated scale effects appear in the bottom boundary conditions of the flow velocity and sediment fall velocity. However, the scale effect would be relatively modest. The Reynolds effect becomes more significant for erosional or mixed erosional/depositional turbidity currents as Reynolds-mediated scale effects also appear in the sediment entrainment relation. Numerical simulations have been conducted at three different scales by considering densimetric Froude scaling alone as well as combined densimetric Froude and Reynolds similarity. Simulation results confirm that although the scaling of densimetric Froude number alone can produce scale-invariable results for conservative density currents, variations occur in the case of turbidity currents. The results become scale invariant when densimetric Froude and Reynolds similarities are satisfied simultaneously.

NHWAVE: Consistent boundary conditions and turbulence modeling

Large-scale σ-coordinate ocean circulation models neglect the horizontal variation of σ in the calculation of stress terms and boundary conditions. Following this practice, the effects of surface and bottom slopes in the dynamic surface and bottom boundary conditions have been usually neglected in the available non-hydrostatic wave-resolving models using a terrain-following grid. In this paper, we derive consistent surface and bottom boundary conditions for the normal and tangential stress fields as well as a Neumann-type boundary condition for scalar fluxes. Further, we examine the role of surface slopes in the predicted near-surface velocity and turbulence fields in surface gravity waves. By comparing the predicted velocity field in a deep-water standing wave in a closed basin, we show that the consistent boundary conditions do not generate unphysical vorticity at the free surface, in contrast to commonly used, simplified stress boundary conditions developed by ignoring all contributions except vertical shear in the transformation of stress terms. In addition, it is shown that the consistent boundary conditions significantly improve predicted wave shape, velocity and turbulence fields in regular surf zone breaking waves, compared with the simplified case. A more extensive model-data comparison of various breaking wave properties in different types of surface breaking waves is presented in companion papers (Derakhti et al., 2016a,b).

Curvature Approximation Effects in the Free Vibration Analysis of Functionally Graded Shells

The present work investigates the effects of the curvature terms in the three-dimensional (3D) equilibrium equations used for the free vibration analysis of functionally graded material (FGM) structures. The 3D equilibrium equations have been written in general orthogonal curvilinear coordinates which are valid for spherical shells. They automatically degenerate in those for cylindrical shells and plates considering one of the two radii of curvature and both radii of curvature equal to infinite, respectively. The approximation of curvature terms in the 3D equilibrium equations has been evaluated by means of frequency analyses. Results obtained via 3D equilibrium equations with exact geometry have been compared with those calculated via 3D equilibrium equations written with the approximation of the curvature terms. The effects of the curvature approximations depend on the thickness and curvature of the structures, on the materials, lamination sequences and FGM laws, on the frequency orders and vibration modes. The resulting system of second order partial differential equations has been reduced into a system of first order partial differential equations redoubling the variables. Therefore, the exponential matrix method has been employed using a layer wise approach. The final 3D equations have been solved in exact form considering harmonic displacement components and simply supported structures. The approximation of the curvature terms has been introduced in the 3D equilibrium shell equations. For numerical reasons, interlaminar continuity conditions and the top and bottom boundary and loading conditions have been written including the exact geometry. The introduction of curvature approximations only in the equilibrium equations is sufficient to obtain an exhaustive qualitative analysis of the importance of curvature terms in the free vibration problems for FGM structures.

Simulated Asian-Australian monsoon with a spectral element atmospheric general circulation model

A low-top version of SEMANS (Spectral Element Model with Atmospheric Near Space resolved) has been used to carry out numerical simulation on characteristics of Asian- Australian Monsoon (A-AM) in the work. The simulation results are validated with ERA- Interim reanalysis dataset and precipitation data from satellite remote sensing. It's shown that the model can reproduce the major climatic features of A-AM with stronger easterly in the tropical Eastern Pacific, and a weaker northerly component in the Northern Hemisphere.The simulated precipitation rate is larger and the double ITCZ (Inter-Tropical Convergence Zone) in the tropical Eastern Pacific in the northern spring is not reproduced. A due to the absence of variation longer than a year in the bottom boundary conditions, the model cannot reproduce the relationships between the monsoon indexes and the surface air temperature in the broad area near the equator.

Nonhydrostatic Quasi-3D Model Coupled with the Dynamic Rough Wall Law for Simulating Flow over a Rough Bed with Submerged Boulders

AbstractA reliable depth-integrated model that is suited for capturing the complex flow found with an array of submerged boulders over a permeable rough bed can be an attractive tool for river management. This paper presents a new nonhydrostatic quasi-three-dimensional (3D) calculation method called the general bottom velocity computation of the fourth-degree polynomial equation for vertical distribution of velocity coupled with the dynamic wall law (GBVC4-DWL). The computational domain of the GBVC4-DWL method considers the presence of the boulders, a vortex layer on the bed surface, and a roughness layer under the bed surface. The novelty of this study is that it introduces a dynamic rough wall law for the bottom boundary condition that includes the continuity and momentum equations for vortex and roughness layers to evaluate the nonequilibrium flow induced by the boulders. Comparisons of the results of the present method with previous experimental results and results of previous calculation methods indi...

An analytical model for contaminant transport in landfill composite liners considering coupled effect of consolidation, diffusion, and degradation.

One-dimensional mathematical model is developed to investigate the behavior of contaminant transport in landfill composite liner system considering coupled effect of consolidation, diffusion, and degradation. The first- and second-type bottom boundary conditions are used to derive the steady-state and quasi-steady-state analytical solutions. The concentration profiles obtained by the proposed analytical solution are in good agreement with those obtained by the laboratory tests. The bottom concentration and flux of the soil liners can be greatly reduced when the degradation effect and porosity changing are considered. For the case under steady-state, the bottom flux and concentration for the case with t 1/2 =10years can be 2.8 and 5.5 times lower than those of the case with t 1/2 =100years, respectively. The bottom concentration and flux of the soil liners can be greatly reduced when the coefficient of volume compressibility decreases. For quasi-steady-state and with t 1/2 = 10years, the bottom flux and concentration for the case with m v = 0.02/MPa can be 17.4 and 21 times lower than the case with m v = 0.5/MPa. This may be due to the fact that the true fluid velocity induced by consolidation is greater for the case with high coefficient of volume compressibility. The bottom flux for the case with single compacted clay liner (CCL) can be 1.5 times larger than that for the case with GMB/CCL considering diffusion and consolidation for DCM. The proposed analytical model can be used for verification of more complicated numerical models and assessment of the coupled effect of diffusion, consolidation, and degradation on contaminant transport in landfill liner systems.