Numerical Reference Solutions Research Articles

Influence of groundwater flow in fractured aquifers on standing column wells performance

A coupled model of a standing column well is developed to evaluate the influence of groundwater flow in fractured aquifers. Heat transfer and groundwater flow within a standing column well and its surrounding ground is modeled by means of a resistance and capacity network. The work demonstrates that the presence of a single fracture zone embedded within the bedrock has a significant beneficial effect on the performance of standing column wells. Temperature differences between the homogeneous and fractured aquifer were particularly important when the well was operated at typical bleed ratios. The results show that the model developed in this paper is in good agreement with numerical reference solutions.

An extension of the SIMPLE based discontinuous Galerkin solver to unsteady incompressible flows

SummaryIn this paper, we present a SIMPLE based algorithm in the context of the discontinuous Galerkin method for unsteady incompressible flows. Time discretization is done fully implicit using backward differentiation formulae (BDF) of varying order from 1 to 4. We show that the original equation for the pressure correction can be modified by using an equivalent operator stemming from the symmetric interior penalty (SIP) method leading to a reduced stencil size.To assess the accuracy as well as the stability and the performance of the scheme, three different test cases are carried out: the Taylor vortex flow, the Orr‐Sommerfeld stability problem for plane Poiseuille flow and the flow past a square cylinder. (1) Simulating the Taylor vortex flow, we verify the temporal accuracy for the different BDF schemes. Using the mixed‐order formulation, a spatial convergence study yields convergence rates of k + 1 and k in the L2‐norm for velocity and pressure, respectively. For the equal‐order formulation, we obtain approximately the same convergence rates, while the absolute error is smaller. (2) The stability of our method is examined by simulating the Orr–Sommerfeld stability problem. Using the mixed‐order formulation and adjusting the penalty parameter of the symmetric interior penalty method for the discretization of the viscous part, we can demonstrate the long‐term stability of the algorithm. Using pressure stabilization the equal‐order formulation is stable without changing the penalty parameter. (3) Finally, the results for the flow past a square cylinder show excellent agreement with numerical reference solutions as well as experiments. Copyright © 2015 John Wiley & Sons, Ltd.

A Numerically Efficient Damping Model for Acoustic Resonances in Microfluidic Cavities

Acoustofluidic damping is a crucial factor that limits the attainable acoustic amplitudes and therefore the effectiveness of acoustofluidic devices. It can be traced back to viscous and thermal dissipation in the bulk and in the boundary layers at cavity walls or suspended particles. However, numerical 3D simulations that include all relevant physics are prohibitively expensive since the acoustic boundary layers need to be resolved. We present a way to incorporate the dissipation effects into a synthetic acoustofluidic loss factor for the use in 3D device simulations. It comes at minimum numerical cost since boundary layers are resolved analytically. Our results and the validity of the physical assumptions we make in the derivation have been verified by analytical and numerical reference solutions. The acoustofluidic loss factor is easily incorporated in device models for a numerically feasible and quantitatively accurate prediction of acoustic amplitudes. In this sense, our work represents the missing link that allows to make not only qualitative but also quantitative predictions of acoustofluidic forces in realistic 3D devices.

Free vibration analysis of homogeneous piezoelectric structures using specific hexahedral elements with rotational DOFs

This paper deals with the free vibration validation of a family of mechanical and piezoelectric eight-node hexahedral elements, presenting rotational degrees of freedom in addition to the classical displacement ones. Their formulations are based on the 3D virtual rotations of a nodal fiber within the finite element that enriches the mechanical displacement vector, stiffness and mass matrices approximations. The performances of these elements for free vibration problems are assessed by means of several beam, plate and shell modal tests, and their results are compared with analytical, experimental and numerical reference solutions. In particular, it is shown that these elements are well adapted for evaluating the modal response of thin and thick elastic and piezoelectric structures, and their accuracy is close to those of the classical mechanical and piezoelectric 20-node hexahedral elements.

Thermal resistance and capacity model for standing column wells operating under a bleed control

A fully coupled multiphysics model involving heat transfer and groundwater flow within a standing column well and its surrounding ground was modeled by means of a thermal resistance and capacity network coupled to an analytical solution. The transient groundwater velocity field and aquifer drawdown are addressed by applying a temporal superposition technique to the so-called Theis analytical equation. The heat pumps are integrated into the model, thereby allowing the effect of its entering water temperature on its capacity and coefficient of performance to be accounted for. To increase the flexibility of the approach, a three-level bleed control and an on-off sequence is included in the model, in order to allow the simulation of the dynamics of a system operation. The results show that the model developed in this paper is consistent with numerical reference solutions.

Lagrangian Stochastic Modelling of Dispersion in the Convective Boundary Layer with Skewed Turbulence Conditions and a Vertical Density Gradient: Formulation and Implementation in the FLEXPART Model

A correction for the vertical gradient of air density has been incorporated into a skewed probability density function formulation for turbulence in the convective boundary layer. The related drift term for Lagrangian stochastic dispersion modelling has been derived based on the well-mixed condition. Furthermore, the formulation has been extended to include unsteady turbulence statistics and the related additional component of the drift term obtained. These formulations are an extension of the drift formulation reported by Luhar et al. (Atmos Environ 30:1407–1418, 1996) following the well-mixed condition proposed by Thomson (J Fluid Mech 180:529–556, 1987). Comprehensive tests were carried out to validate the formulations including consistency between forward and backward simulations and preservation of a well-mixed state with unsteady conditions. The stationary state CBL drift term with density correction was incorporated into the FLEXPART and FLEXPART-WRF Lagrangian models, and included the use of an ad hoc transition function that modulates the third moment of the vertical velocity based on stability parameters. Due to the current implementation of the FLEXPART models, only a steady-state horizontally homogeneous drift term could be included. To avoid numerical instability, in the presence of non-stationary and horizontally inhomogeneous conditions, a re-initialization procedure for particle velocity was used. The criteria for re-initialization and resulting errors were assessed for the case of non-stationary conditions by comparing a reference numerical solution in simplified unsteady conditions, obtained using the non-stationary drift term, and a solution based on the steady drift with re-initialization. Two examples of “real-world” numerical simulations were performed under different convective conditions to demonstrate the effect of the vertical gradient in density on the particle dispersion in the CBL.

Modelling of transient heat conduction with diffuse interface methods

We present a survey on different numerical interpolation schemes used for two-phase transient heat conduction problems in the context of interface capturing phase-field methods. Examples are general transport problems in the context of diffuse interface methods with a non-equal heat conductivity in normal and tangential directions to the interface. We extend the tonsorial approach recently published by Nicoli M et al (2011 Phys. Rev. E 84 1–6) to the general three-dimensional (3D) transient evolution equations. Validations for one-dimensional, two-dimensional and 3D transient test cases are provided, and the results are in good agreement with analytical and numerical reference solutions.

An Improved FE-BE Method for Solving the Fluid-Structure Interaction Problems of Plates

This paper proposed a numerical approach called improved FE-BE (finite element-boundary element) method to solve the fluid-structure interaction problems of plate-like systems. In order to avoid the mesh of both faces of the thin plates, some dumb structures were added to make the plate systems closed. Thus the conventional FE-BE method can be adopted to solve this problem. The dynamic response equation of the inner region can be obtained by the FE method, and the acoustic added mass and damping coefficients of the exterior region can be obtained by the BE method. Then the final fluid-structure interaction equation can be easily solved. Numerical results of some examples are computed to demonstrate the validation of the present method. The comparison of numerical results and reference solutions shows that the proposed method is acceptable for solving the fluid-structure interaction of plates.

A gas-kinetic model for 2D transcritical shallow water flows propagating over dry bed

The solution of fluid dynamics problems has recently witnessed a remarkable increase in the formulation of Boltzmann-based approaches, mainly triggered by the numerous advantages stemming from the concrete possibility of employing a low number of allowed velocities, which is the basis of the Lattice-Boltzmann (LB) methods. The field of Shallow Water (SW) modeling also took advantage of these techniques, but there are still open problems related to the practical impossibility to simulate transcritical flows while retaining the intrinsic simplicity of the LB approaches. This problem is even more crucial if one considers that transcritical flows always develop whenever a transition over dry bed occurs. Since such vertically integrated models are currently the mostly employed ones for simulating technically interesting flows, and since these flows often require the flooding of an initially dry bed, the application of LB methods seems to be facing its limits. The Gas Kinetic Method (GKM) overcomes this issue, integrating the Boltzmann equations in continuous velocity space. In this work, we formulate and extensively test a GKM-based model for solving the SW equations, which is able to manage the propagation over uneven dry bed. The benchmarking, carried out against analytical, experimental and previously proposed numerical reference solutions, shows promising results of the proposed approach.

Efficient meshless SPH method for the numerical modeling of thick shell structures undergoing large deformations

The objective of this paper is to present an extension of the Lagrangian Smoothed Particle Hydrodynamics (SPH) method to solve three-dimensional shell-like structures undergoing large deformations. The present method is an enhancement of the classical stabilized SPH commonly used for 3D continua, by introducing a Reissner–Mindlin shell formulation, allowing the modeling of moderately thin structure using only one layer of particles in the shell mid-surface. The proposed Shell-based SPH method is efficient and very fast compared to the classical continuum SPH method. The Total Lagrangian Formulation valid for large deformations is adopted using a strong formulation of the differential equilibrium equations based on the principle of collocation. The resulting non-linear dynamic problem is solved incrementally using the explicit time integration scheme, suited to highly dynamic applications. To validate the reliability and accuracy of the proposed Shell-based SPH method in solving shell-like structure problems, several numerical applications including geometrically non-linear behavior are performed and the results are compared with analytical solutions when available and also with numerical reference solutions available in the literature or obtained using the Finite Element method by means of ABAQUS© commercial software.

A new class of Moving-Least-Squares WENO–SPH schemes

We present a new class of meshless Lagrangian particle methods based on the SPH formulation of Vila and Ben Moussa, combined with a new weighted essentially non-oscillatory (WENO) reconstruction technique on moving point clouds in multiple space dimensions. The key idea is to produce for each particle first a set of high order accurate Moving-Least-Squares (MLS) reconstructions on a set of different reconstruction stencils. Then, these reconstructions are combined with each other using a non-linear WENO technique in order to capture at the same time discontinuities and to maintain accuracy and low numerical dissipation in smooth regions. The numerical fluxes between interacting particles are subsequently evaluated using this MLS–WENO reconstruction at the midpoint between two particles, in combination with a Riemann solver that provides the necessary stabilization of the scheme based on the underlying physics of the governing equations. We propose the use of two different Riemann solvers: the Rusanov flux and an Osher-type flux. The use of monotone fluxes together with a WENO reconstruction ensures accuracy, stability, robustness and an essentially non-oscillatory solution without the artificial viscosity term usually employed in conventional SPH schemes.To our knowledge, this is the first time that the WENO method, which has originally been developed for mesh-based schemes in the Eulerian framework on fixed grids, is extended to meshfree Lagrangian particle methods like SPH in multiple space dimensions.We test the new algorithm on two dimensional blast wave problems and on the classical one-dimensional Sod shock tube problem for the Euler equations of compressible gas dynamics. We obtain a good agreement with the exact or numerical reference solution in all cases and an improved accuracy and robustness compared to existing standard SPH schemes.

Application of the meshless finite volume particle method to flow-induced motion of a rigid body

We present a new approach to numerical modelling of incompressible flow of fluid about an elastically mounted rigid structure with large body motions. The solution is based on the Finite Volume Particle Method (FVPM), a meshless generalisation of the mesh-based finite volume method. The finite volume particles are allowed to overlap, without explicit connectivity, and can therefore move arbitrarily to follow the motion of a wall. Here, FVPM is employed with a pressure projection method for fully incompressible flow coupled with motion of a rigid body. The developed extension is validated for Vortex-Induced Vibration (VIV) of a circular cylinder in laminar crossflow. To minimise computational effort, non-uniform particle size and arbitrary Lagrangian–Eulerian particle motion schemes are employed, with radial basis functions used to define the particle motion near the cylinder. Close agreement is demonstrated between the FVPM results and a reference numerical solution. Results confirm the feasibility of FVPM as a new approach to the modelling of flow with strongly coupled rigid-body dynamics.

Viscous bubbly flows simulation with an interface SPH model

The multi-fluid SPH formulation by Grenier et al. (2009) is extended to study practical problems where bubbly flows play an important role for production processes of the offshore industry. Since these flows are dominated by viscous and surface tension effects, the proposed formulation includes specific models of these physical effects and validations on specific benchmark test cases are carefully performed. The numerical outputs are validated against analytical and numerical reference solutions, and accuracy and convergence of the proposed numerical model are assessed. This model is then used to simulate viscous incompressible bubbly flows of increasing complexity. These flows include the evolution of isolated bubbles, the merging of two bubbles, and the separation process in a bubbly flow. In each case, results are compared to reference solutions and the influence of the Bond number on these interfacial flow evolutions is investigated in detail.

Time-domain numerical simulations of multiple scattering to extract elastic effective wavenumbers

Elastic wave propagation is studied in a heterogeneous two-dimensional medium consisting of an elastic matrix containing randomly distributed circular elastic inclusions. The aim of this study is to determine the effective wavenumbers when the incident wavelength is similar to the radius of the inclusions. A purely numerical methodology is presented, with which the limitations usually associated with low scatterer concentrations can be avoided. The elastodynamic equations are integrated by a fourth-order time-domain numerical scheme. An immersed interface method is used to accurately discretize the interfaces on a Cartesian grid. The effective field is extracted from the simulated data, and signal-processing tools are used to obtain the complex effective wavenumbers. The numerical reference solution thus obtained can be used to check the validity of multiple scattering analytical models. The method is applied to the case of concrete. A parametric study is performed on longitudinal and transverse incident plane waves at various scatterer concentrations. The phase velocities and attenuations determined numerically are compared with predictions obtained with multiple scattering models, such as the Independent Scattering Approximation model, the Waterman–Truell model, and the more recent Conoir–Norris model.

3-D seakeeping analysis with water on deck and slamming. Part 1: Numerical solver

A three-dimensional seakeeping numerical solver is developed to handle occurrence and effects of water-on-deck and bottom slamming. It couples (A) the rigid-ship motions with (B) the water flowing along the deck and (C) bottom slamming events. Problem A is studied with a 3-D weakly nonlinear potential flow solver based on the weak-scatterer hypothesis. Problem B, and so local and global induced green-water loads, are investigated by assuming shallow-water conditions onto the deck. Problem C is examined through a Wagner-type wedge-impact analysis. Within the coupling between A and B: the external seakeeping problem furnishes the initial and boundary conditions to the in-deck solver in terms of water level and velocity along the deck profile; in return the shallow-water problem makes available to the seakeeping solver the green-water loads to be introduced as additional loads into the rigid-motion equations. Within the coupling between A and C: the instantaneous ship configuration and its kinematic and dynamic conditions with respect to the incident waves will fix the parameters for the local impact problem; in return the slamming and water-entry pressures are integrated in the vessel region of interest and introduced as additional loads into the rigid-motion equations. The resulting numerical solver can study efficiently the ship interaction with regular and irregular sea states and the forward motion with limited speed of the vessel. This is crucial to perform reliable and feasible statistical investigations of vessel behavior. Main elements of the solver are described and validated against reference numerical solutions and model tests.

The singular edge-based smoothed finite element method for stationary dynamic crack problems in 2D elastic solids

In this paper, the recently developed singular edge-based smoothed finite element method (sES-FEM) is further developed for dynamic crack analysis in two-dimensional elastic solids. The objective of this work is to provide an efficient and accurate numerical simulation tool for the dynamic fracture behaviors of linear elastic solids in the framework of the strain smoothing approaches. Following this approach, the strains are smoothed and the system stiffness matrix is thus performed using the strain smoothing technique over the smoothing domains associated with the element edges. In order to accurately capture the singular fields at the crack-tip, a two-layer singular 5-node crack-tip element is employed. The governing dynamic equations are transformed into a weakened weak (W2) form, which is then discretized into a sES-FEM system of time-dependent equations to be solved by the unconditionally stable implicit Newmark time integration method. To analyze the fracture behaviors of linear elastic solids, mixed-mode dynamic stress intensity factors (DSIFs) are evaluated using the domain forms of the interaction integrals in terms of the smoothing technique. Four test examples including pure mode-I and mixed-modes are studied to validate the accuracy of the proposed method. The computed results for the normalized DSIFs are compared with analytical and other numerical reference solutions in a wide range of benchmark dynamic crack problems which shows high accuracy of the sES-FEM.

Development of a three-dimensional piezoelectric isogeometric finite element for smart structure applications

In this paper, a 3D electromechanical-coupled isogeometric finite element is presented. An important advantage of such elements is the possibility to incorporate geometrical data of computer-aided design systems into the description of the finite element models. In this work, the isogeometric approach is extended to a coupled electromechanical finite element with mechanical and electrical degrees of freedom. The proposed piezoelectric finite element is based on nonuniform rational B-spline shape functions. In order to demonstrate the functionality and the advantages of the isogeometric piezoelectric element for smart structure and structural health monitoring applications, various examples are presented. The results obtained with isogeometric elements agree very well with analytical and numerical reference solutions. It is shown that the better geometrical approximation introduced by isogeometric elements results in an improvement of the solution quality.

JOSEPHINE: A parallel SPH code for free-surface flows

JOSEPHINE is a parallel Smoothed Particle Hydrodynamics program, designed to solve unsteady free-surface flows. The adopted numerical scheme is efficient and has been validated on a first case, where a liquid drop is stretched over the time. Boundary conditions can also be modelled, as it is demonstrated in a second case: the collapse of a water column. Results show good agreement with both reference numerical solutions and experiments. The use of parallelism allows significant reduction of the computational time, even more with large number of particles. JOSEPHINE has been written so that any untrained developers can handle it easily and implement new features. Program summaryProgram title:JOSEPHINECatalogue identifier: AELV_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AELV_v1_0.htmlProgram obtainable from: CPC Program Library, Queenʼs University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 5139No. of bytes in distributed program, including test data, etc.: 22 833Distribution format: tar.gzProgramming language: Fortran 90 and OpenMPIComputer: All shared or distributed memory parallel processors, tested on a Xeon W3520, 2.67 GHz.Operating system: Any system with a Fortran 90 compiler and MPI, tested on Debian Linux.Has the code been vectorised or parallelised?: The code has been parallelised but has not been explicitly vectorised.RAM: Dependent upon the number of particles.Classification: 4.12Nature of problem:JOSEPHINE is designed to solve unsteady incompressible flows with a free-surface and large deformations.Solution method:JOSEPHINE is an implementation of Smoothed Particle Hydrodynamics. SPH is a Lagrangian mesh free particle method, thus, no explicit tracking procedure is required to catch the free surface. Incompressibility is satisfied using a weakly compressible model. Boundary conditions at walls are enforced by means of the ghost particles technique. The free-surface dynamic and kinematic conditions are applied implicitly.Running time: 15 mn on 4 processors for the dam-break case with 5000 particles, dependent upon the real duration (2 s here).

VALIDATION AND APPLICATION OF AN HIGH-ORDER SPECTRAL DIFFERENCE METHOD FOR FLOW INDUCED NOISE SIMULATION

The main goal of this paper is to develop an efficient numerical algorithm to compute the radiated far field noise provided by an unsteady flow field from bodies in arbitrary motion. The method computes a turbulent flow field in the near fields using a high-order spectral difference method coupled with large-eddy simulation approach. The unsteady equations are solved by advancing in time using a second-order backward difference formulae scheme. The nonlinear algebraic system arising from the time discretization is solved with the nonlinear lower–upper symmetric Gauss–Seidel algorithm. In the second step, the method calculates the far field sound pressure based on the acoustic source information provided by the first step simulation. The method is based on the Ffowcs Williams–Hawkings approach, which provides noise contributions for monopole, dipole and quadrupole acoustic sources. This paper will focus on the validation and assessment of this hybrid approach using different test cases. The test cases used are: a laminar flow over a two-dimensional (2D) open cavity at Re = 1.5 × 103 and M = 0.15 and a laminar flow past a 2D square cylinder at Re = 200 and M = 0.5. In order to show the application of the numerical method in industrial cases and to assess its capability for sound field simulation, a three-dimensional turbulent flow in a muffler at Re = 4.665 × 104 and M = 0.05 has been chosen as a third test case. The flow results show good agreement with numerical and experimental reference solutions. Comparison of the computed noise results with those of reference solutions also shows that the numerical approach predicts noise accurately.

Displacement discontinuity method for modeling axisymmetric cracks in an elastic half-space

This paper describes a displacement discontinuity method for modeling axisymmetric cracks in an elastic half-space or full space. The formulation is based on hypersingular integral equations that relate displacement jumps and tractions along the crack. The integral kernels, which represent stress influence functions for ring dislocation dipoles, are derived from available axisymmetric dislocation solutions. The crack is discretized into constant-strength displacement discontinuity elements, where each element represents a slice of a cone. The influence integrals are evaluated using a combination of numerical integration and a recursive procedure that allows for explicit integration of hyper- and Cauchy singularities. The accuracy of the solution at the crack tip is ensured by adding corrective stresses across the tip element. The method is validated by a comparison with analytical and numerical reference solutions.