Far-field Boundary Conditions Research Articles

Entropy stable far-field boundary conditions for the compressible Navier-Stokes equations

We consider external compressible-flow problems with open boundaries. We begin by presenting the basic idea, namely the introduction of an infinite buffer zone around the computational domain, for the advection-diffusion equation. We discretise the problem with a summation-by-parts finite-volume scheme and prove convergence; a key step is the renormalisation of the solution. Next, we make the analogous rescaling of the entropy function for the Navier-Stokes equations and prove that a finite-volume scheme is entropy stable.Finally, we present numerical computations with a vortex governed by the Navier-Stokes equations interacting with the buffer zone. As proven, the scheme is stable, and with a sufficient number of grid points and with some numerical diffusion in the buffer zone, the reflections are modest.

Exact domain truncation for the Morse-ingard equations

Morse and Ingard [1] give a coupled system of time-harmonic equations for the temperature and pressure of an excited gas. These equations form a critical aspect of modeling trace gas sensors. Like other wave propagation problems, the computational problem must be closed with suitable far-field boundary conditions. Working in a scattered-field formulation, we adapt a nonlocal boundary condition proposed in [2] for the Helmholtz equation to this coupled system. This boundary condition uses a Green's formula for the true solution on the boundary, giving rise to a nonlocal perturbation of standard transmission boundary conditions. However, the boundary condition is exact and so Galerkin discretization of the resulting problem converges to the restriction of the exact solution to the computational domain. Numerical results demonstrate that accuracy can be obtained on relatively coarse meshes on small computational domains, and the resulting algebraic systems may be solved by GMRES using the local part of the operator as an effective preconditioner. These numerical results taken together combine several advanced techniques, including higher-order finite elements, geometric multigrid in curvilinear geometry, native use of complex arithmetic, and incorporation of nonlocal operators. These are tied together in a high-level simulation using the Firedrake library.

A multiscale analytical-numerical method for the coupled heat and mass transfer in the extended meniscus region considering thin-film evaporation in microchannels

A Computational Fluid Dynamics (CFD) framework is established to investigate the microscale transport phenomena in the evaporating extended meniscus region formed in rectangular microchannels. A wide range of microchannel widths (W = 1 to 250 µm) and wall superheats (ΔT = 0.01 to 5.0 K) are considered. Prior to performing CFD simulations, it is first necessary to employ thin-film evaporation modeling based on an augmented Young–Laplace model and kinetic-theory based model for the evaporating mass flux across curved surfaces to find the exact shape of the liquid-vapor interface along with disjoining and capillary pressure distributions. Two widely-used methods for the determination of boundary conditions (BCs) at the beginning of the thin-film region (i.e., setting δ0″ = 0 and finding δ0′ by the far-field BC or setting δ0′ = 0 and finding δ0″ by the far-field BC), where δ is film thickness, are thoroughly examined and the first approach is recommended because of its better convergence and ease of implementation in the thin-film evaporation model. Then, the two-dimensional domain of the extended meniscus is generated and discretized to simulate the evaporating liquid flow through the UDF programming in ANSYS Fluent. The developed framework is shown to be a simple yet powerful and practical method for accurately predicting the evaporation mass and heat fluxes from extended menisci in microchannels. The CFD simulation results indicate that the previous one-dimensional models which represent the state of the art in the literature in the field are not able to predict the rate of evaporative heat transfer from the extended meniscus in microchannels with an acceptable degree of accuracy. Based on the CFD simulations results, a multiple regression analysis is employed to establish several simple thermal resistance correlations. The correlations can be straightforwardly integrated into macroscale mathematical models of micro and miniature heat pipes for analyzing their thermal characteristics.

A stable scheme of the Curvilinear Shallow Water Equations with no-penetration and far-field boundary conditions

This paper presents a stable and highly accurate numerical tool for computing river flows in urban areas, which is a first step towards a numerical tool for flood predictions. We start with the (linearized) well-posedness analysis by Ghader and Nordström (2014), where far-field boundary conditions were proposed and extend their analysis to include wall boundaries. Specifically, we employed high-order Summation-by-parts (SBP) finite-difference operators to construct a scheme for the Shallow Water Equations. We also developed a stable SBP scheme with Simultaneous Approximation Terms that impose far-field and wall boundaries. Finally, we extended the schemes and their stability proofs to non-Cartesian domains. To demonstrate the strength of the schemes, we performed computations for problems with exact solutions to obtain second, third, and fourth (2, 3, 4) convergence rates. Finally, we applied the 4th-order scheme to steady river channels, the canal (or flood-control channel simulations), and dam-break problems. The results show that the imposition of the boundary conditions is stable, and the far-field boundaries cause no visible reflections at the boundaries.

Overview of Selected Serpentine Duct Simulations Using Computational Fluid Dynamics

Computational fluid dynamics (CFD) was used to perform simulations of multiple serpentine inlet ducts for comparison to experimental data. The simulations included two baseline duct geometries configured to match their respective wind tunnel tests. Each configuration was tested and simulated with and without flow control vanes. Some of the simulations included the geometric detail of the instrumentation probes at the aerodynamic interface plane. This paper highlights selected issues that impact robustness and accuracy. The topics include far-field boundary conditions and various methods of modeling the instrumentation probes. This paper summarizes discussion topics and recommendations for CFD practitioners preparing to simulate similar cases.

A kinetic scheme with variable velocities and relative entropy

A new kinetic model is proposed where the equilibrium distribution with bounded support has a range of velocities about two average velocities in 1D. In 2D, the equilibrium distribution function has a range of velocities about four average velocities, one in each quadrant. In the associated finite volume scheme, the average velocities are used to enforce the Rankine–Hugoniot jump conditions for the numerical diffusion at cell-interfaces, thereby capturing steady discontinuities exactly. The variable range of velocities is used to provide additional diffusion in smooth regions. Further, a novel kinetic theory based expression for relative entropy is presented which, along with an additional criterion, is used to identify expansions and smooth flow regions. Appropriate flow tangency and far-field boundary conditions are formulated for the proposed kinetic model. Several benchmark 1D and 2D compressible flow test cases are solved to demonstrate the efficacy of the proposed solver.

Numerical investigation of solitary wave breaking over a slope based on multi-phase smoothed particle hydrodynamics

This study presents a numerical investigation of the solitary wave breaking over a slope by using the multi-phase smoothed particle hydrodynamics (SPH) method. Four different computational models are proposed to solve the gas-related far-field boundary conditions, and the model with the least disturbance to the internal flow field is selected. Since the artificial viscous coefficient can greatly affect the wave-breaking location, an empirical equation is fitted to quickly determine the optimal value of the artificial viscous coefficient. In addition, the turbulence model and three-dimensional effect on the wave breaking are discussed in this study. The results show that the present two-dimensional multi-phase SPH without a turbulence model can capture the macroscopic characteristics of the flow before the vortices convert to three dimensional flow structures caused by the wave breaking. Then, the processes of shoaling solitary wave breaking with different slopes and relative wave heights are simulated. Compared with the single-phase SPH, the multi-phase SPH is of great help in improving the prediction of wave breaking. A vortex similar to the Rankine Vortex is observed near the wave crest. Its intensity affects the pressure distribution of the gas, and its relative position to the wave crest is relevant to the energy transfer from the water to the gas. During the solitary wave propagating from deep water to shallow water, energy dissipation of gas and water shows four different stages. In the stage of energy dissipation, the gas can absorb the great energy from the water, which effectively dissipates the wave energy.

Fluid-Structure Interaction: Impact of Reservoir Simulation Approach Considering Far-Field Boundary Condition in Dam Seismic Response

Fluid-Structure Interaction: Impact of Reservoir Simulation Approach Considering Far-Field Boundary Condition in Dam Seismic Response

Influence of Shape Factor and Non-Linear Stretching of the Bullet-Shaped Object on the Mixed Convection Boundary Layer Flow and Heat Transfer with Viscous Dissipation and Internal Heat Generation

The two-dimensional incompressible axisymmetric mixed convection magnetohydrodynamic fluid flow and energy transfer over a bullet-shaped object with a non-linear stretching surface have been investigated. The main goal of this problem is to discuss the effect of the shape and size of the bullet-shaped object on the fluid velocity and temperature distributions. The present analysis has been performed in about two cases ε=0.0 and 2.0. Therefore, fluid velocity and temperature distributions have been investigated in two types of flow geometries such as the thicker surface (s ≥ 2) and the thinner surface (0 < s < 2) of the bullet-shaped object. The equations for momentum and heat transfer have been converted into ODEs by using suitable local similarity transformations. These equations have been performed with a recently developed spectral quasi-linearization method (SQLM). This method helps to identify the accuracy, validity, and convergence of the present solution. The novelty of the present work has been applying the recently developed numerical method to solve these highly nonlinear differential equations. The investigation shows that in the case of a thicker bullet-shaped object (s ≥ 2) the velocity and temperature profiles do not converse the far-field boundary condition asymptotically but cross the axis with an upright angle and the boundary layer structure has no definite shape whereas in the case of a thinner bullet-shaped object (0 < s < 2) the velocity profile converge the ambient condition asymptotically and the boundary layer structure has a definite shape. The innovation of this current work lies in the unification of relevant physical parameters into the governing equations and trying to explain how the flow properties are affected by these parameters.

Numerical study of freestream turbulence effects on the nose cones in supersonic flow

PurposeThis paper aims to numerically investigate the influence of shock wave and freestream turbulence interaction on the parabolic and spherically blunted nose cones at supersonic speed.Design/methodology/approachUsing density-based solver, the three-dimensional steady-state simulation is carried out. The working fluid is calorically perfect that obeys ideal gas law and the no-slip boundary conditionis given to the surface of the nose cone. Pressure far-field boundary condition is imposed at the boundary of the computational domain by giving freestream Mach number, freestream static pressure and temperature.FindingsThe growth rate of the boundary layer is faster on the spherically blunted nose cone, hence, the overall drag force is higher than the parabolic nose cone. Temperature at the edge of the boundary layer is increased due to the early ampli-fication of instabilities by the upstream disturbance. In this sense, the effects of freestream turbulence depend on its level, freestream conditions, strength and type of shock wave and zone of influence.Research limitations/implicationsSimulations are carried out for the flow Mach number 2.0 at zero angles of attack for the freestream conditions of the flow at an altitude of 10,000 m.Practical implicationsThe phenomenon of shock wave–turbulence interaction occurs in flow regimes from transonic to hypersonic speeds and finds a wide range of applications, especially in the design of aircraft and missiles configurations.Originality/valueThe phenomenon of compression wave and freestream turbulence interaction around the commonly used nose cones in the case of aircraft, missiles, etc., is investigated. The performance characteristics such as aerodynamic drag, boundary layer dynamics and the nature of flow around the different nose cones at zero angle of attack are illustrated.

The Tectonic Map and Structural Provinces of the Late Neoproterozoic Egyptian Nubian Shield: Implications for Crustal Growth of the Arabian–Nubian Shield (East African Orogen)

The Late Neoproterozoic Egyptian Nubian Shield (ENS) has attracted increasing attention since the establishment of the Egyptian Geological Survey and Mining Authority (EGSMA (1896)), which conducted the first mapping. In the last three decades, rapid improvements in analytical techniques, along with field-oriented studies, have made it possible for the interpretation of the ENS as an integral part of the juvenile Arabian–Nubian Shield (ANS) as a portion of the East African Orogen (EAO). However, a consistent tectonic map of the ENS does not exist. Presentation of such a map is the main objective of the present work where Landsat-based lithological discrimination is combined with the systematic structural investigation and careful reassessment of previously published geological maps. Our interpretation of this map indicates that in accordance with previous work, the basement units of the Eastern Desert (ED) can be divided into three structural domains: Northern, Central, and South-Eastern Desert provinces. The proposed provinces are ascribed to three different plate tectonic far-field boundary conditions. The earliest magmatic, metamorphic, and tectonic history was set off by the approximate north–south convergence of the Gabgaba-Gebeit, Jiddah-Asir terranes, and Eastern Desert-Midyan terranes along the Yanbo-Onib-Sol-Hamid-Gerf-Allaqi-Heiani (YOSHGAH) suture between c. 800 and 620 Ma. The second event between c. 640 and 580 Ma was correlated with the Nabitah Orogeny when the Ad Dawadimi-Ar Rayn terranes in the eastern Arabian Shield accreted to the earlier consolidated arc terranes (Afif and Tathlith terranes). During this period, east–west convergence between northwestern Saudi Arabia and the Central Eastern Desert (CED) in Egypt caused a pronounced strike-slip deformation associated with the displacement along the wider Najd Fault system. The third orogenic phase, which was younger than c. 580 Ma, was controlled by the evolution and retreat of the Cadomian Arc. The northern portions of the ENS and the northern Arabian Shield experienced extension whereas extensive post-orogenic magmatism was related to mantle delamination and associated crustal thinning.

On the lifetimes of two-dimensional droplets on smooth wetting patterns

We study the evolution and lifetime of droplets evaporating on a smooth chemical pattern, which is characterised by a spatially varying contact angle. We formulate a model that combines the evaporation rate of the droplet for a given volume with the static stability of the droplet as the volume changes in time quasi-statically. We derive an exact equation for the evaporation rate that is studied analytically under the limiting cases of nearly neutral wetting and highly hydrophilic conditions. We find that the evaporation rate of the droplet is highly dependent on the size and shape of the fictitious infinity where a far-field boundary conditions needs to be applied. We also study how the droplet’s lifetime depends on the averaged contact angle and strength varepsilon of the chemical pattern, observing that the lifetime of the droplet is maximised for a droplet with average contact angle pi /2 and with varepsilon lesssim 0.1.

Accurate and Efficient Finite Difference Method for the Black–Scholes Model with No Far-Field Boundary Conditions

Accurate and Efficient Finite Difference Method for the Black–Scholes Model with No Far-Field Boundary Conditions

Crack field analysis by optical DIC of short cracks in Zircaloy-4

The prediction of fatigue life is most difficult for short cracks, as their local conditions may differ from what is predicted using the remote applied loading and crack geometry. Digital image correlation (DIC) can be utilized to analyze images from an optical microscope (OM), which facilitates the local characterization of the crack field. This paper presents a Finite-Element-based approach that uses DIC-obtained displacement data to retrieve the crack field and quantify the local crack driving force. With the assumption of linear anisotropic elasticity, the change in both the Mode I and Mode II crack intensity factors over a fatigue cycle can be extracted using the interaction integral method. This allows the determination of the local driving force for short crack propagation. The application of this method is demonstrated by the full-field analysis of short fatigue cracks in blocky alpha Zircaloy-4. The sensitivity of the analysis is investigated to factors that include DIC subset size, uncertainties in crack tip position and lower quality data in the crack vicinity. This technique requires no prior knowledge of theoretical solutions or far-field boundary conditions, and it can be applied to the tip of a tortuous crack by defining an appropriate local frame of reference. The analysis is applied here to cracks under similar mode I loading that propagate on the prism plane at different rates in directions that are parallel and perpendicular to the c-axis of the hexagonal unit cell.

On the Determination of Thermal Conductivity From Frequency Domain Thermoreflectance Experiments

Abstract The Fourier and the hyperbolic heat conduction equations were solved numerically to simulate a frequency-domain thermoreflectance (FDTR) experiment. Numerical solutions enable isolation of pump and probe laser spot size effects and use of realistic boundary conditions. The equations were solved in time domain and the phase lag between the temperature of the transducer (averaged over the probe laser spot) and the modulated pump laser signal was computed for a modulation frequency range of 200 kHz–200 MHz. Numerical calculations showed that extracted values of the thermal conductivity are sensitive to both the pump and probe laser spot sizes, while analytical solutions (based on Hankel transform) cannot isolate the two effects. However, for the same effective (combined) spot size, the two solutions are found to be in excellent agreement. If the substrate (computational domain) is sufficiently large, the far-field boundary conditions were found to have no effect on the computed phase lag. The interface conductance between the transducer and the substrate was found to have some effect on the extracted thermal conductivity. The hyperbolic heat conduction equation yielded almost the same results as the Fourier heat conduction equation for the particular case studied. The numerically extracted thermal conductivity value (best fit) for the silicon substrate considered in this study was found to be about 82–108 W/m/K, depending on the pump and probe laser spot sizes used.

A peridynamic model for crevice corrosion damage

A new peridynamic (PD) model for crevice corrosion damage is introduced and cross-validated with experimental results from the literature. Electrochemical mechanisms are reduced to a simple metal-ion concentration-dependent corrosion rate. Computations over domains with extreme aspect ratios like those seen in crevices are efficiently performed by modifying the PD formulation to accommodate arbitrary-shape horizons. The model is validated against experimental results on bolted washers. Simulations predict both the location and the depth of deep corrosion trenches that form at a distance from the crevice mouth. Far-field boundary conditions are used to autonomously reproduce the experimentally observed progression of crevice corrosion. The PD model results also show that the two factors considered here (diffusion-driven transport of dissolved metal ions, and dissolution being dependent on their concentration) are sufficient to predict the evolution of crevice corrosion.

Study on the maximum tangential strain criterion for the initiation of the wing-crack under uniaxial compression

The global stress field function of an infinite plate with a closed inclined crack under uniaxial compression is presented, then the stress field at the crack tip is obtained, namely the formulae of KI, KII and T-stress are obtained. A new maximum tangential strain (MTSN) criterion is proposed to predict the initiation of the crack under compression by considering singular and non-singular terms based on the classical MTSN criterion. The influence factors on KII are analyzed and the effects of T-stress on the tangential strain at the crack tip under different critical plastic zones (rc), dip angel of crack (β), friction coefficients of crack surface (f) and Poisson’s ratio (ν) are studied. The comparison of the wing-crack initiation angle measured from the experimental results and predicted by the proposed criterion and other researchers’ models are made. The following findings are obtained. First of all, the proposed global stress field function satisfies the far-field and crack surface stress boundary conditions. The singular term of KII and non-singular term of T-stress vary with the effective shear stress along the crack surface. When the effective shear stress is zero, there is still singularity at the crack tip, which is completely opposite to the classical theory, and there are two components of T-stress (i.e. Tx and Ty, which is parallel and perpendicular to the crack surface respectively). While there is a third component of T-stress (i.e. Txy, which along the xy direction) when the effective shear stress is greater than zero. Next, the comparison of the wing-crack initiation angle measured from the experimental results and predicted by the proposed criterion and other researchers’ models are made. The results of the proposed method are consistent with the experiments very well, especially when the effective shear stress decreases to zero with the increase of friction coefficient (f). Finally, the tangential strain field at the crack tip are effected significantly by the critical plastic zones (rc), dip angel of crack (β), friction coefficients of crack surface (f) and Poisson’s ratio (ν).

Dynamic Assessment of Basin-scale Pressure Dissipation and Impact on Storage Capacity Using Accelerated Methodology

Storage capacity of closed or semi-closed aquifer systems is constrained by build-up of overpressure. Estimation of pressure- constrained storage capacity of a basin-scale resource should be obtained by dynamic simulation of multi-site CO2 injection. The limitation on pressure can occur near the injection where overpressure is greatest, or further away where the basin is shallower. Pressure build-up and dissipation can be impacted by heterogeneity and far-field boundary conditions, which may involve significant uncertainty. The impact of uncertainty in heterogeneity on pressure-constrained capacity is not very well understood due to the computational burden of basin-scale simulations. In this study, we quantify the uncertainty of storage capacity due multi- site injection within a large-scale heterogeneous aquifer system using ensemble-based methods. The approach is accelerated by employing simplified vertical equilibrium methods for the forward model. We find that uncertainty in large-scale heterogeneity results in a wide distribution of estimated capacity, however the estimated mean is less sensitive to variation of the geostatistical model. The effect of average permeability and change in boundary conditions has a greater impact on the estimated mean than variation in the underlying geostatistical model. This implies that for large-scale systems, heterogeneity is important to take into account, but precision in the geostatistical model is less important than other basin-scale quantities.

On the far-field boundary condition treatment in the framework of aeromechanical computations using ANSYS CFX

This numerical study aims at predicting the reflective behavior of different conventional inlet and outlet far-field boundary conditions as well as available non-reflecting boundary conditions (NRBC) implemented in the commercial CFD solver ANSYS CFX. An isolated rotor model of an axial turbine stage with prescribed blade displacement is applied as test case to consider a representative application case, while at the same time provoke an unsteady flow field featuring pronounced flow perturbations in the far-field. Since the reflective behavior of the implemented boundary conditions was found inadequate in the given application case, a zonal treatment of the inlet and outlet far-field, based on a modification of the governing Navier-Stokes equations, is investigated. The applied approach has proven its capability to suppress spurious reflections reliably, while at the same time ensures a preservation of the reference flow conditions within the required domain extensions. The results of a case study considering calculation domains of different spatial extent and different treatments of their respective far-fields suggest variations in the steady flow aerodynamics to be of moderate influence on the predicted aerodynamic damping, while spurious reflections were found to falsify the unsteady aerodynamics considerably.

Time limit for using the semi-infinite heat transfer solutions: a novel approach

In the present study, authors are conceptually showing that the classically considered time limit to use the semi-infinite approximate solutions is highly conservative, particularly at the internal location(s) inside the finite heat conduction medium. Accordingly, a new length scale, which accounts the heat propagation from the far-field boundary condition as well, is proposed to ascertain the prolonged time limit. The proposed time limit is obtained by comparing the temperature distribution in a finite heat conduction problem with its equivalent semi-infinite model. Overall, three standard one-dimensional heat conduction problems are analysed and the proposed time limit is found to be valid in all three problems. The new time limit will certainly boost the utility of the semi-infinite solutions and rejuvenate the interest of the scientific community in such solutions.