Numerical Boundary Conditions Research Articles

High order compact augmented methods for Stokes equations with different boundary conditions

This paper is devoted to fourth order compact schemes and fast algorithms for solving stationary Stokes equations with different boundary conditions numerically. One of the main ideas is to decouple the Stokes equations into three Poisson equations for the pressure and the velocity via the pressure Poisson equation (PPE). The augmented strategy is utilized to provide numerical boundary conditions for the pressure. Different velocity boundary conditions require different interpolation strategies for the augmented methods. The augmented variable is solved by the GMRES method. A new simple and efficient preconditioning strategy has also been developed to accelerate the convergence of the GMRES iteration. Numerical examples presented in this paper confirmed the designed convergence order and the efficiency of the new methods.

Cut-PFEM: a Particle Finite Element Method using unfitted boundary meshes

AbstractIn this work, we present a novel unfitted mesh boundary strategy in the context of the Particle Finite Flement Method (PFEM) aiming to improve endemic limitations of the PFEM relative to boundary conditions treatment and mass conservation. In this new methodology, which we called Cut-PFEM, the fluid–wall interaction is not performed by adding interface elements, as is done in the standard PFEM boundaries. Instead, we use an implicit representation of (all or some of) the boundaries by introducing the use of a level set function. Such distance function detects the elements trespassing the (virtual) contours of the domain to equip them with opportunely boundary conditions, which are variationally enforced using Nitsche’s method. The proposed Cut-PFEM circumvents important issues associated with the standard PFEM contact detection algorithm, such as the artificial addition of mass to the computational domain and the anticipation of contact time. Furthermore, the Cut-PFEM represents a natural ground for the imposition of alternative wall boundary conditions (e.g., pure slip) which pose significant difficulties in a standard PFEM framework. Several numerical examples, featuring both no-slip and slip boundary conditions, are presented to prove the accuracy and robustness of the method in two-dimensional and three-dimensional scenarios.

Simulating the Photospheric to Coronal Plasma Using Magnetohydrodynamic Characteristics. I. Data-driven Boundary Conditions

We develop a general description of how information propagates through a magnetohydrodynamic (MHD) system based on the method of characteristics and use that to formulate numerical boundary conditions that are intrinsically consistent with the MHD equations. Our formulation includes two major advances for simulations of the Sun. First, we derive data-driven boundary conditions that optimally match the state of the plasma inferred from a time series of observations of a boundary (e.g., the solar photosphere). Second, our method directly handles random noise and systematic bias in the observations, and finds a solution for the boundary evolution that is strictly consistent with MHD and maximally consistent with the observations. We validate the method against a Ground Truth (GT) simulation of an expanding spheromak. The data-driven simulation can reproduce the GT simulation above the photosphere with high fidelity when driven at high cadence. Errors progressively increase for lower driving cadence until a threshold cadence is reached and the driven simulation can no longer accurately reproduce the GT simulation. However, our characteristic formulation of the boundary conditions still requires adherence of the boundary evolution to the MHD equations even when the driven solution departs from the true solution in the driving layer. That increasing departure clearly indicates when additional information at the boundary is needed to fully specify the correct evolution of the system. The method functions even when no information about the evolution of some variables on the lower boundary is available, albeit with a further decrease in fidelity.

Simulation of the dendrite growth during directional solidification under steady magnetic field using three-dimensional cellular automaton method coupled with Eulerian multiphase

A three-dimensional cellular automaton method coupled with Eulerian multiphase is developed to investigate the dendrite growth during directional solidification under a steady magnetic field. In this model, the heat and solute transports, dendrite growth, thermal and solutal buoyancy, Thermo-Electro-Magnetic force and induced Lorentz force are taken into account. The simulation results reveal that asymmetric characteristics of dendrites are observed during the directional solidification of Al-2.5wt%Cu alloy under a steady magnetic field because of the intense melt flow occurring in the vicinity of the solid-liquid interface. The simulated results show that the numerical boundary condition has significant influence on the formation of tilted interface due to the changing of melt flow. Meanwhile, dendrites with various of preferred orientation are set to investigate the melt flow on the evolution of dendritic morphology during the directional solidification under the steady magnetic field. Finally, a criterion of u→l∇cl in the vicinity of solid-liquid interface is proposed to illustrate the evolution mechanism of dendritic morphology under steady magnetic field, i.e., the region with u→l∇cl>0 promoting the dendrites growth and u→l∇cl<0 suppressing the dendrites growth.

Evaluation of Turbulence Models in Unsteady Separation

Unsteady separation is a phenomenon that occurs in many flows and results in increased drag, decreased lift, noise emission, and loss of efficiency or failure in flow devices. Turbulence models for the steady or unsteady Reynolds-averaged Navier–Stokes equations (RANS and URANS, respectively) are commonly used in industry; however, their performance is often unsatisfactory. The comparison of RANS results with experimental data does not clearly isolate the modeling errors, since differences with the data may be due to a combination of modeling and numerical errors, and also to possible differences in the boundary conditions. In the present study, we use high-fidelity large-eddy simulation (LES) results to carry out a consistent evaluation of the turbulence models. By using the same numerical scheme and boundary conditions as the LES, and a grid on which grid convergence was achieved, we can isolate modeling errors. The calculations (both LES and RANS) are carried out using a well-validated, second-order-accurate code. Separation is generated by imposing a freestream velocity distribution, that is modulated in time. We examined three frequencies (a rapid, flutter-like oscillation, an intermediate one in which the forcing and the flow have the same timescales, and a quasi-steady one). We also considered three different pressure distributions, one with alternating favorable and adverse pressure gradients (FPGs and APGs, respectively), one oscillating between an APG and a zero-pressure gradient (ZPG), and one with an oscillating APG. All turbulence models capture the general features of this complex unsteady flow as well or better than in similar steady cases. The presence, during the cycle, of times in which the freestream pressure-gradient is close to zero affects significantly the model performance. Comparing our results with those in the literature indicates that numerical errors due to the type of discretization and the grid resolution are as significant as those due to the turbulence model.

Tamed stability of finite difference schemes for the transport equation on the half-line

In this paper, we prove that, under precise spectral assumptions, some finite difference approximations of scalar leftgoing transport equations on the positive half-line with numerical boundary conditions are ℓ 1 \ell ^1 -stable but ℓ q \ell ^q -unstable for any q &gt; 1 q&gt;1 . The proof relies on the accurate description of the Green’s function for a particular family of finite rank perturbations of Toeplitz operators whose essential spectrum belongs to the closed unit disk and with a simple eigenvalue of modulus 1 1 embedded into the essential spectrum.

Numerical Study of Hypersonic Near-Continuum Flow by Improved Slip Boundary Condition

Appropriate numerical models and boundary conditions are required to accurately model hypersonic flows involving local nonequilibrium effects, which can be induced by the wall effect and large gradients of flow properties in the flowfield. Considering the strong density gradient near the wall for hypersonic flows, an improved Gökçen slip model is proposed in this paper by introducing a scaled mean free path to the conventional Gökçen’s model. Hypersonic argon flows over a circular cylinder and a blunt plate in the near-continuum regime are simulated by the proposed slip model with the conventional Navier–Stokes–Fourier equations and an extended continuum model, which was recently developed to model the shear nonequilibrium effect in high-speed flows. The accuracy of the new boundary model is evaluated by comparing the results with the direct simulation Monte Carlo method. The effects of Knudsen number, Mach number, and wall temperature on the nonequilibrium degree and aerodynamic predictions are also investigated. The numerical results show that the improved model can well predict the surface friction and heat transfer coefficients, exhibiting much better performance than the existing continuum models.

Implementing the direct relaxation process in the stochastic particle method for flexible molecular collisions

Multi-scale phenomena are prevalent and significant across various disciplines. For multi-scale flow physics in the gas-kinetic theory based on Boltzmann equation or its simplified mathematical models (called Boltzmann model equations), the multi-scale mechanism can be modeled by the philosophy of unified modeling, where the free transport behaviors of gas particles and their collision behaviors are coupled by the temporal integral solutions (or characteristic line solutions) of Boltzmann model equations, which leads to a unified/multi-scale property in all scales. Also, the stochastic particle methods are based on these Boltzmann model equations. The corresponding numerical methods are, thus, limited by these model equations. This paper aims to overcome this restriction by replacing these modeled collision operators with a simple direct relaxation (DR) process. Since the collision term of Boltzmann model equation should fulfill the correct relaxation rates of non-equilibrium macro-variables, such as stress tensor and heat flux vector, along with other basic properties, such as conservation and H theorem, the DR process is designed to be directly based on these crucial relaxation rates. Therefore, with the DR strategy for calculating particle collisions, the numerical method can be established without constructing collision operator. Furthermore, the DR has the flexibility and simplicity to recover various models. In this work, Xu's and Yuan's new models are recovered in to illustrate the validation and performance of DR. Moreover, since at the inlet/outlet boundaries, subsonic, supersonic, and hypersonic flows can simultaneously exist, a generalized numerical boundary condition is also considered in the particle methods developed in this paper. Finally, the validation and accuracy of the present method are examined with a series of test cases.

A high-resolution scheme for axisymmetric hydrodynamics based on the 2D GRP solvers

In the simulation of axisymmetric fluids, imposing an appropriate numerical boundary condition at the symmetrical axis r=0 is an inevitable difficulty. To maintaining the conservation property, we choose to keep its form coincided with the numerical scheme on interior control volumes. Then we obtain necessary interface values in both scenarios from solving the two-dimensional generalized Riemann problems (2D GRPs) and their one-sided versions. Therein the effects of transversal variation and geometrical source are specifically emphasized to exhibit the genuine multi-dimensionality of the entire algorithm. In the construction of the one-sided 2D GRP solver, we have to face the difficulty brought by the singularity of the source at r=0. The ingenious combination of acoustic approximation, symmetry argument, and L’Hospital’s rule is proposed in obtaining a pivotal limiting value there. Several challenging tests are provided to demonstrate the effectiveness and robustness of our approach.

On the stability of totally upwind schemes for the hyperbolic initial boundary value problem

Abstract In this paper, we present a numerical strategy to check the strong stability (or GKS-stability) of one-step explicit totally upwind schemes in 1D with numerical boundary conditions. The underlying approximated continuous problem is the one-dimensional advection equation. The strong stability is studied using the Kreiss–Lopatinskii theory. We introduce a new tool, the intrinsic Kreiss–Lopatinskii determinant, which possesses remarkable regularity properties. By applying standard results of complex analysis, we are able to relate the strong stability of numerical schemes to the computation of a winding number, which is robust and cheap. The study is illustrated with the Beam–Warming scheme together with the simplified inverse Lax–Wendroff procedure at the boundary.

A fast numerical method with non-iterative source term for pseudo-two-dimension lithium-ion battery model

In this work, a fast numerical method with non-iterative volumetric source for pseudo-two-dimensional (P2D) model is proposed for lithium-ion batteries and has been implemented in the open-source software OpenFOAM. The direct determination of the precise potential distribution is enabled by correct boundary conditions derived from the Butler-Volmer equation and charge conservation, due to the mathematical characteristic of the Laplace equation and the small grid scale involved. This method eliminates repeated iterations to obtain the accurate volumetric source distribution for each step required by the sequential full-order model (SFOM) method. The numerical solving boundary conditions coincide with those of the conventional P2D model in terms of physical meaning. Nevertheless, some innovations were introduced to render it more computationally efficient. We integrated this method into OpenFOAM, and applied it to typical examples with the same parameters as COMSOL Multiphysics, the output results were found to match almost completely with the latter. This method holds tremendous promise for practical application in battery design. It accurately represents chemical processes within the battery, including mass and charge transfer, migration and diffusion. Additionally, it improves efficiency over the full-order P2D model.

3D modelling of surface spreading and underground dam groundwater recharge: Egri Creek Subbasin, Turkey.

This study investigated surface spreading and underground dam recharge methods to replenish groundwater in Turkey's Egri Creek Sub-basin of the Kucuk Menderes River Basin. A three-dimensional numerical model was employed for this purpose. Field and lab data are provided to the model for realistic simulations. Pumping test results were used to determine the aquifer parameters. The laboratory works involved sieve analysis, permeability tests, and porosity and water content prediction. The numerical model's boundary conditions were determined from the geological and hydrogeological characteristics of the study area. Initial conditions were expressed regarding water content and pressure head in the vadose zone. The numerical model was satisfactorily validated by simulating water levels in three different pumping wells in the study area. Seven different scenarios, each having a different pool size, were investigated for the surface spreading recharge method. The results showed that a pool size of 30 × 30m with a 6-m depth basin was the most optimal choice, raising the groundwater level to about 29.3m. On the other hand, it was found that an underground dam could raise the levels by an average of 9.5m, which might not be significant to warrant the construction.

Field synergy and thermodynamic evaluation of a mini-duct with several protrusions utilizing the cross-flow method: A numerical exercise

The field synergy and entropy generation analysis of turbulent flow in a mini rectangular duct featuring surface protrusions has been numerically explored in a finite volume approach. The governing differential is solved iteratively using adequate numerical boundary conditions and a second-order upwind technique. In the post-processing, entropy generation, and irreversibilities are measured. The duct Reynolds number (ReDh, duct) has been varied from 17,831 to 53,492 at a fixed nozzle Reynolds number (ReDh, nz) of 5104. The effects of the duct Reynolds number on the thermal and frictional irreversibility have been quantified. Thermal, frictional, and total irreversibilities have been reported to rise with ReDh, duct. Furthermore, it is discovered that thermal irreversibility dominates over frictional irreversibility. It has been found that conducting triangular protrusions creates more entropy than conducting rectangular protrusions. Filed synergy analysis has also been carried out by integrating the energy equation. Heat transfer rate has been seen to increase with synergy angle (α). Triangular protrusions have a greater field synergy angle (α) than rectangular protrusions.

Jacobian-free Newton–Krylov method for the simulation of non-thermal plasma discharges with high-order time integration and physics-based preconditioning

A preconditioning framework for the numerical simulation of non-thermal streamer discharges is developed using the Jacobian-free Newton-Krylov (JFNK) method. A reduced plasma fluid model is considered, consisting of electrons, one positive ion, one negative ion, and the electrostatic potential. The plasma kinetics model includes ionization, electron-ion recombination, electron attachment, electron detachment, and ion-ion recombination. The governing equations are made dimensionless, discretized in space with finite differences, and integrated in time with a fully implicit method based on high-order backward differentiation formulas. The preconditioning framework is based on a linearized form of the governing equations and physics-based operator splitting. The efficiency of the preconditioning strategy is assessed through two test cases: streamer propagation between parallel plates and an axisymmetric pin-to-pin discharge. The fully implicit approach overcomes traditional restrictions in the time step size due to processes such as electron drift, electron diffusion, and dielectric relaxation. Excellent performance is observed through relevant statistics of the JFNK solver, although the number of linear iterations increases for the pin-to-pin discharge when nonlinear numerical boundary conditions are imposed at the electrodes. Performance studies show scalability with O(100-1000) processors for O(10M) unknowns with ample room for optimization.

Reconstruction of proper numerical inlet boundary conditions for draft tube flow simulations using machine learning

This paper discusses an innovative strategy to determine appropriate mean and fluctuating inlet boundary conditions for the numerical flow simulations of a hydraulic turbine draft tube. Precise and comprehensive measurements at a draft tube inlet are hard to perform and thus rarely available. Moreover, its complex nature and strong sensitivity to imposed inlet boundary conditions render this flow particularly challenging to predict with numerical methods. The innovative strategy uses machine learning to properly reconstruct the incomplete or unknown inlet boundary conditions from available downstream flow information. Reynolds-Averaged Navier Stokes (RANS) and Large-Eddy Simulations (LES) turbulence modeling methods are used to investigate the draft tube and the numerical results are compared to experimental measurements. Reference simulations are first conducted with standard inlet boundary conditions commonly used in previous numerical works. For RANS, the innovative strategy significantly improves the mean velocity profiles and static pressure evolution inside the draft tube. For LES, its use with an artificial upstream extension allows a proper reconstruction of the upstream turbulent field and a more precise and realistic description of the draft tube head losses.

Response Analysis of Curved Tunnel under Near-Field Long-Period Ground Motion Considering Seismic Wave Propagation Effect

In this paper, long-period ground motion is used as the dynamic input to study the performance evolution of curved tunnel lining structure under seismic wave propagation excitation. This paper presents numerical studies on seismic waves, considering propagation effect, and aims to illustrate the response principle and structural failure mechanism of tunnel structures under long-period ground motion. Firstly, based on the dynamic analysis method, the dynamic balance equation of a tunnel under the seismic wave effect was analyzed. Secondly, this equation was applied to the 3D finite element software, the corresponding numerical model and boundary conditions were established, and the parameterized numerical analysis of the tunnel was carried out. Finally, according to the numerical simulation results, the seismic response principle and structural failure mechanism of a tunnel structure under long-period ground motion were discussed. The research results show that the depth and segment thickness of the tunnel significantly affect the seismic performance of the tunnel. The seismic response mechanism of a curved tunnel is complex, which shows that the relative displacements on the left and right symmetrical positions are different. The displacement inside the curve is less than the displacement outside the curve. Compared with other types of ground motion, the near-site motion considering the seismic wave propagation effect can lead to large deformation of the tunnel, which damages the lining structure greatly, and the enhancement effect is prominent for the long shield tunnel.

Long waves with a small amplitude on the surface of the water behave dynamically in nonlinear lattices on a non-dimensional grid

Approximation and analysis are used for investigating accurate soliton solutions of the ill-posed Boussinesq (IPB) equation. The investigated model explains shallow-water gravitational waves. It examines one-dimensional nonlinear strings and lattices. IPB explains small-amplitude surface waves on nonlinear strings and lattices. We provide unique analytical solutions to analyze numerical beginning and boundary conditions. A solution’s quality is judged by its divergence from analytical predictions. Physical wave properties are illustrated.

Transparent boundary condition for simulating rogue wave solutions in the nonlinear Schrödinger equation.

This paper addresses the construction of numerical boundary conditions for simulating rogue wave solutions in the nonlinear Schrödinger equation. While three kinds of commonly used boundary conditions require a big enough computational domain to reproduce solutions faithfully in the central domain, we propose transparent boundary conditions for the Peregrine soliton and Kuznetsov-Ma breather solutions, respectively. For both solutions, these boundary conditions require a smaller computational domain than other boundary conditions to attain the best accuracy of the Crank-Nicolson scheme and selected mesh size, which will be referred to as the "acceptable accuracy" below. In particular, the computational domain with these boundary conditions is only 1/16 as small as others in the simulations of the Peregrine soliton solution. As a result, they reduce both the memory requirement and the computing time for the Peregrine soliton solution.

Internal absorbing boundary conditions for closed-aperture wavefield decomposition in solid media with unknown interiors.

We present a method to create an internal numerical absorbing boundary within elastic solid media whose properties are largely unknown and use it to create the first wavefield separation method that retrieves all orders of outgoing elastic wavefield constituents for real data recorded on a closed free surface. The recorded data are injected into a numerical finite-difference (FD) simulation along a closed, transparent surface, and the new internal numerical absorbing boundary condition achieves high attenuation of the ingoing waves radiated from the injection surface. This internal wave absorption enables the data injection to radiate all outgoing waves for experimental domains that include arbitrary unknown scatterers in the interior. The injection-absorption-based separation scheme is validated using three-dimensional (3D) synthetic modeling and a real data experiment acquired using a 3D laser Doppler vibrometer on a granite rock. The wavefield separation method forms a key component of an elastic immersive wave experimentation laboratory, and the ability to numerically absorb ingoing scattered energy in an uncharacterized medium while still radiating the true outgoing energy is intriguing and may lead to other development and applications in the future.

LABYRINTH SEAL CFD CALCULATION AND TEMPERATURE MEASUREMENT INVESTIGATION

This article presents currently obtained results from CFD analysis of the labyrinth seals of an aircraft turbine engine. The process of describing a geometry, grid for numerical calculation and boundary conditions are described. Numerical simulations were performed for the assumed boundary conditions. The presented results show total temperature differences in labyrinth seals compared to published results. An experimental verification of the CFD analysis was also performed to clarify the numerical simulation results. It was based on the labyrinth seal measurement stand. The final part of this study is dedicated to the discussion and the following possible activities on this topic.